author paulson Mon, 29 Apr 2019 16:50:20 +0100 changeset 70212 e886b58bf698 parent 70196 b7ef9090feed child 70213 ff8386fc191d
full proof of algebraic closure, by Paulo de Vilhena
```--- a/src/HOL/Algebra/Algebraic_Closure.thy	Fri Apr 26 16:51:40 2019 +0100
+++ b/src/HOL/Algebra/Algebraic_Closure.thy	Mon Apr 29 16:50:20 2019 +0100
@@ -5,7 +5,7 @@
*)

theory Algebraic_Closure
-  imports Indexed_Polynomials Polynomial_Divisibility Pred_Zorn Finite_Extensions
+  imports Indexed_Polynomials Polynomial_Divisibility Finite_Extensions

begin

@@ -21,18 +21,19 @@
\<lparr> mult := (\<lambda>a \<in> carrier R. \<lambda>b \<in> carrier R. a \<otimes>\<^bsub>R\<^esub> b),
add := (\<lambda>a \<in> carrier R. \<lambda>b \<in> carrier R. a \<oplus>\<^bsub>R\<^esub> b) \<rparr>"

-definition (in ring) \<sigma> :: "'a list \<Rightarrow> (('a list multiset) \<Rightarrow> 'a) list"
+definition (in ring) \<sigma> :: "'a list \<Rightarrow> ((('a list \<times> nat) multiset) \<Rightarrow> 'a) list"
where "\<sigma> P = map indexed_const P"

-definition (in ring) extensions :: "(('a list multiset) \<Rightarrow> 'a) ring set"
+definition (in ring) extensions :: "((('a list \<times> nat) multiset) \<Rightarrow> 'a) ring set"
where "extensions \<equiv> { L \<comment> \<open>such that\<close>.
\<comment> \<open>i\<close>   (field L) \<and>
\<comment> \<open>ii\<close>  (indexed_const \<in> ring_hom R L) \<and>
\<comment> \<open>iii\<close> (\<forall>\<P> \<in> carrier L. carrier_coeff \<P>) \<and>
-           \<comment> \<open>iv\<close>  (\<forall>\<P> \<in> carrier L. \<forall>P \<in> carrier (poly_ring R).
-                       \<not> index_free \<P> P \<longrightarrow> \<X>\<^bsub>P\<^esub> \<in> carrier L \<and> (ring.eval L) (\<sigma> P) \<X>\<^bsub>P\<^esub> = \<zero>\<^bsub>L\<^esub>) }"
+           \<comment> \<open>iv\<close>  (\<forall>\<P> \<in> carrier L. \<forall>P \<in> carrier (poly_ring R). \<forall>i.
+                       \<not> index_free \<P> (P, i) \<longrightarrow>
+                         \<X>\<^bsub>(P, i)\<^esub> \<in> carrier L \<and> (ring.eval L) (\<sigma> P) \<X>\<^bsub>(P, i)\<^esub> = \<zero>\<^bsub>L\<^esub>) }"

-abbreviation (in ring) restrict_extensions :: "(('a list multiset) \<Rightarrow> 'a) ring set" ("\<S>")
+abbreviation (in ring) restrict_extensions :: "((('a list \<times> nat) multiset) \<Rightarrow> 'a) ring set" ("\<S>")
where "\<S> \<equiv> law_restrict ` extensions"

@@ -65,7 +66,7 @@
lemma (in field) law_restrict_is_field: "field (law_restrict R)"
proof -
have "comm_monoid_axioms (law_restrict R)"
-    using m_comm unfolding comm_monoid_axioms_def law_restrict_carrier law_restrict_mult by auto
+    using m_comm unfolding comm_monoid_axioms_def law_restrict_carrier law_restrict_mult by auto
then interpret L: cring "law_restrict R"
using cring.intro law_restrict_is_ring comm_monoid.intro ring.is_monoid by auto
have "Units R = Units (law_restrict R)"
@@ -73,7 +74,7 @@
thus ?thesis
using L.cring_fieldI unfolding field_Units law_restrict_carrier law_restrict_zero by simp
qed
-
+
lemma law_restrict_iso_imp_eq:
assumes "id \<in> ring_iso (law_restrict A) (law_restrict B)" and "ring A" and "ring B"
shows "law_restrict A = law_restrict B"
@@ -112,38 +113,38 @@

subsection \<open>Partial Order\<close>

-lemma iso_incl_backwards:
+lemma iso_incl_backwards:
assumes "A \<lesssim> B" shows "id \<in> ring_hom A B"
using assms by cases

-lemma iso_incl_antisym_aux:
+lemma iso_incl_antisym_aux:
assumes "A \<lesssim> B" and "B \<lesssim> A" shows "id \<in> ring_iso A B"
-proof -
-  have hom: "id \<in> ring_hom A B" "id \<in> ring_hom B A"
+proof -
+  have hom: "id \<in> ring_hom A B" "id \<in> ring_hom B A"
using assms(1-2)[THEN iso_incl_backwards] by auto
-  thus ?thesis
+  thus ?thesis
using hom[THEN ring_hom_memE(1)] by (auto simp add: ring_iso_def bij_betw_def inj_on_def)
qed

-lemma iso_incl_refl: "A \<lesssim> A"
+lemma iso_incl_refl: "A \<lesssim> A"
by (rule iso_inclI[OF ring_hom_memI], auto)

-lemma iso_incl_trans:
+lemma iso_incl_trans:
assumes "A \<lesssim> B" and "B \<lesssim> C" shows "A \<lesssim> C"
using ring_hom_trans[OF assms[THEN iso_incl_backwards]] by auto

lemma (in ring) iso_incl_antisym:
assumes "A \<in> \<S>" "B \<in> \<S>" and "A \<lesssim> B" "B \<lesssim> A" shows "A = B"
-proof -
-  obtain A' B' :: "('a list multiset \<Rightarrow> 'a) ring"
+proof -
+  obtain A' B' :: "(('a list \<times> nat) multiset \<Rightarrow> 'a) ring"
where A: "A = law_restrict A'" "ring A'" and B: "B = law_restrict B'" "ring B'"
using assms(1-2) field.is_ring by (auto simp add: extensions_def)
-  thus ?thesis
+  thus ?thesis
using law_restrict_iso_imp_eq iso_incl_antisym_aux[OF assms(3-4)] by simp
qed

-lemma (in ring) iso_incl_partial_order: "partial_order_on \<S> (rel_of (\<lesssim>) \<S>)"
-  using iso_incl_refl iso_incl_trans iso_incl_antisym by (rule partial_order_on_rel_ofI)
+lemma (in ring) iso_incl_partial_order: "partial_order_on \<S> (relation_of (\<lesssim>) \<S>)"
+  using iso_incl_refl iso_incl_trans iso_incl_antisym by (rule partial_order_on_relation_ofI)

lemma iso_inclE:
assumes "ring A" and "ring B" and "A \<lesssim> B" shows "ring_hom_ring A B id"
@@ -174,14 +175,14 @@
show "indexed_const \<in> ring_hom R (image_ring indexed_const R)"
using inj_imp_image_ring_iso[OF indexed_const_inj_on] unfolding ring_iso_def by auto
next
-    fix \<P> :: "('a list multiset) \<Rightarrow> 'a" and P
+    fix \<P> :: "(('a list \<times> nat) multiset) \<Rightarrow> 'a" and P and i
assume "\<P> \<in> carrier (image_ring indexed_const R)"
then obtain k where "k \<in> carrier R" and "\<P> = indexed_const k"
unfolding image_ring_carrier by blast
-    hence "index_free \<P> P" for P
+    hence "index_free \<P> (P, i)" for P i
unfolding index_free_def indexed_const_def by auto
-    thus "\<not> index_free \<P> P \<Longrightarrow> \<X>\<^bsub>P\<^esub> \<in> carrier (image_ring indexed_const R)"
-     and "\<not> index_free \<P> P \<Longrightarrow> ring.eval (image_ring indexed_const R) (\<sigma> P) \<X>\<^bsub>P\<^esub> = \<zero>\<^bsub>image_ring indexed_const R\<^esub>"
+    thus "\<not> index_free \<P> (P, i) \<Longrightarrow> \<X>\<^bsub>(P, i)\<^esub> \<in> carrier (image_ring indexed_const R)"
+     and "\<not> index_free \<P> (P, i) \<Longrightarrow> ring.eval (image_ring indexed_const R) (\<sigma> P) \<X>\<^bsub>(P, i)\<^esub> = \<zero>\<^bsub>image_ring indexed_const R\<^esub>"
by auto
from \<open>k \<in> carrier R\<close> and \<open>\<P> = indexed_const k\<close> show "carrier_coeff \<P>"
unfolding indexed_const_def carrier_coeff_def by auto
@@ -194,7 +195,7 @@
subsection \<open>Chains\<close>

definition union_ring :: "(('a, 'c) ring_scheme) set \<Rightarrow> 'a ring"
-  where "union_ring C =
+  where "union_ring C =
\<lparr> carrier = (\<Union>(carrier ` C)),
monoid.mult = (\<lambda>a b. (monoid.mult (SOME R. R \<in> C \<and> a \<in> carrier R \<and> b \<in> carrier R) a b)),
one = one (SOME R. R \<in> C),
@@ -281,7 +282,7 @@
using field_chain by simp

show "a \<otimes>\<^bsub>union_ring C\<^esub> b \<in> carrier (union_ring C)"
-    using R(1-3) unfolding same_laws(1)[OF R(1-3)] unfolding union_ring_def by auto
+    using R(1-3) unfolding same_laws(1)[OF R(1-3)] unfolding union_ring_def by auto
show "(a \<otimes>\<^bsub>union_ring C\<^esub> b) \<otimes>\<^bsub>union_ring C\<^esub> c = a \<otimes>\<^bsub>union_ring C\<^esub> (b \<otimes>\<^bsub>union_ring C\<^esub> c)"
and "a \<otimes>\<^bsub>union_ring C\<^esub> b = b \<otimes>\<^bsub>union_ring C\<^esub> a"
and "\<one>\<^bsub>union_ring C\<^esub> \<otimes>\<^bsub>union_ring C\<^esub> a = a"
@@ -290,7 +291,7 @@
next
show "\<one>\<^bsub>union_ring C\<^esub> \<in> carrier (union_ring C)"
using ring.ring_simprules(6)[OF ring_chain] assms same_one_same_zero(1)
-    unfolding union_ring_carrier by auto
+    unfolding union_ring_carrier by auto
qed

lemma union_ring_is_abelian_group:
@@ -308,7 +309,7 @@
show "(a \<oplus>\<^bsub>union_ring C\<^esub> b) \<otimes>\<^bsub>union_ring C\<^esub> c = (a \<otimes>\<^bsub>union_ring C\<^esub> c) \<oplus>\<^bsub>union_ring C\<^esub> (b \<otimes>\<^bsub>union_ring C\<^esub> c)"
and "(a \<oplus>\<^bsub>union_ring C\<^esub> b) \<oplus>\<^bsub>union_ring C\<^esub> c = a \<oplus>\<^bsub>union_ring C\<^esub> (b \<oplus>\<^bsub>union_ring C\<^esub> c)"
and "a \<oplus>\<^bsub>union_ring C\<^esub> b = b \<oplus>\<^bsub>union_ring C\<^esub> a"
-   and "\<zero>\<^bsub>union_ring C\<^esub> \<oplus>\<^bsub>union_ring C\<^esub> a = a"
+   and "\<zero>\<^bsub>union_ring C\<^esub> \<oplus>\<^bsub>union_ring C\<^esub> a = a"
using same_one_same_zero[OF R(1)] same_laws[OF R(1)] R(2-4) l_distr a_assoc a_comm by auto
have "\<exists>a' \<in> carrier R. a' \<oplus>\<^bsub>union_ring C\<^esub> a = \<zero>\<^bsub>union_ring C\<^esub>"
using same_laws(2)[OF R(1)] R(2) same_one_same_zero[OF R(1)] by simp
@@ -334,7 +335,7 @@
using field_chain by simp

from \<open>a \<in> carrier R\<close> and \<open>a \<noteq> \<zero>\<^bsub>union_ring C\<^esub>\<close> have "a \<in> Units R"
-      unfolding same_one_same_zero[OF R(1)] field_Units by auto
+      unfolding same_one_same_zero[OF R(1)] field_Units by auto
hence "\<exists>a' \<in> carrier R. a' \<otimes>\<^bsub>union_ring C\<^esub> a = \<one>\<^bsub>union_ring C\<^esub> \<and> a \<otimes>\<^bsub>union_ring C\<^esub> a' = \<one>\<^bsub>union_ring C\<^esub>"
using same_laws[OF R(1)] same_one_same_zero[OF R(1)] R(2) unfolding Units_def by auto
with \<open>R \<in> C\<close> and \<open>a \<in> carrier (union_ring C)\<close> show "a \<in> Units (union_ring C)"
@@ -370,78 +371,84 @@

subsection \<open>Zorn\<close>

+(* ========== *)
+lemma Chains_relation_of:
+  assumes "C \<in> Chains (relation_of P A)" shows "C \<subseteq> A"
+  using assms unfolding Chains_def relation_of_def by auto
+(* ========== *)
+
lemma (in ring) exists_core_chain:
-  assumes "C \<in> Chains (rel_of (\<lesssim>) \<S>)" obtains C' where "C' \<subseteq> extensions" and "C = law_restrict ` C'"
-  using Chains_rel_of[OF assms] by (meson subset_image_iff)
+  assumes "C \<in> Chains (relation_of (\<lesssim>) \<S>)" obtains C' where "C' \<subseteq> extensions" and "C = law_restrict ` C'"
+  using Chains_relation_of[OF assms] by (meson subset_image_iff)

lemma (in ring) core_chain_is_chain:
-  assumes "law_restrict ` C \<in> Chains (rel_of (\<lesssim>) \<S>)" shows "\<And>R S. \<lbrakk> R \<in> C; S \<in> C \<rbrakk> \<Longrightarrow> R \<lesssim> S \<or> S \<lesssim> R"
+  assumes "law_restrict ` C \<in> Chains (relation_of (\<lesssim>) \<S>)" shows "\<And>R S. \<lbrakk> R \<in> C; S \<in> C \<rbrakk> \<Longrightarrow> R \<lesssim> S \<or> S \<lesssim> R"
proof -
fix R S assume "R \<in> C" and "S \<in> C" thus "R \<lesssim> S \<or> S \<lesssim> R"
-    using assms(1) unfolding iso_incl_hom[of R] iso_incl_hom[of S] Chains_def by auto
+    using assms(1) unfolding iso_incl_hom[of R] iso_incl_hom[of S] Chains_def relation_of_def
+    by auto
qed

lemma (in field) exists_maximal_extension:
shows "\<exists>M \<in> \<S>. \<forall>L \<in> \<S>. M \<lesssim> L \<longrightarrow> L = M"
proof (rule predicate_Zorn[OF iso_incl_partial_order])
-  show "\<forall>C \<in> Chains (rel_of (\<lesssim>) \<S>). \<exists>L \<in> \<S>. \<forall>R \<in> C. R \<lesssim> L"
-  proof
-    fix C assume C: "C \<in> Chains (rel_of (\<lesssim>) \<S>)"
-    show "\<exists>L \<in> \<S>. \<forall>R \<in> C. R \<lesssim> L"
-    proof (cases)
-      assume "C = {}" thus ?thesis
-        using extensions_non_empty by auto
-    next
-      assume "C \<noteq> {}"
-      from \<open>C \<in> Chains (rel_of (\<lesssim>) \<S>)\<close>
-      obtain C' where C': "C' \<subseteq> extensions" "C = law_restrict ` C'"
-        using exists_core_chain by auto
-      with \<open>C \<noteq> {}\<close> obtain S where S: "S \<in> C'" and "C' \<noteq> {}"
-        by auto
+  fix C assume C: "C \<in> Chains (relation_of (\<lesssim>) \<S>)"
+  show "\<exists>L \<in> \<S>. \<forall>R \<in> C. R \<lesssim> L"
+  proof (cases)
+    assume "C = {}" thus ?thesis
+      using extensions_non_empty by auto
+  next
+    assume "C \<noteq> {}"
+    from \<open>C \<in> Chains (relation_of (\<lesssim>) \<S>)\<close>
+    obtain C' where C': "C' \<subseteq> extensions" "C = law_restrict ` C'"
+      using exists_core_chain by auto
+    with \<open>C \<noteq> {}\<close> obtain S where S: "S \<in> C'" and "C' \<noteq> {}"
+      by auto

-      have core_chain: "\<And>R. R \<in> C' \<Longrightarrow> field R" "\<And>R S. \<lbrakk> R \<in> C'; S \<in> C' \<rbrakk> \<Longrightarrow> R \<lesssim> S \<or> S \<lesssim> R"
-        using core_chain_is_chain[of C'] C' C unfolding extensions_def by auto
-      from \<open>C' \<noteq> {}\<close> interpret Union: field "union_ring C'"
-          using union_ring_is_field[OF core_chain] C'(1) by blast
+    have core_chain: "\<And>R. R \<in> C' \<Longrightarrow> field R" "\<And>R S. \<lbrakk> R \<in> C'; S \<in> C' \<rbrakk> \<Longrightarrow> R \<lesssim> S \<or> S \<lesssim> R"
+      using core_chain_is_chain[of C'] C' C unfolding extensions_def by auto
+    from \<open>C' \<noteq> {}\<close> interpret Union: field "union_ring C'"
+        using union_ring_is_field[OF core_chain] C'(1) by blast

-      have "union_ring C' \<in> extensions"
-      proof (auto simp add: extensions_def)
-        show "field (union_ring C')"
-          using Union.field_axioms .
-      next
-        from \<open>S \<in> C'\<close> have "indexed_const \<in> ring_hom R S"
-          using C'(1) unfolding extensions_def by auto
-        thus "indexed_const \<in> ring_hom R (union_ring C')"
-          using ring_hom_trans[of _ R S id] union_ring_is_upper_bound[OF core_chain S]
-          unfolding iso_incl.simps by auto
-      next
-        show "a \<in> carrier (union_ring C') \<Longrightarrow> carrier_coeff a" for a
-          using C'(1) unfolding union_ring_carrier extensions_def by auto
-      next
-        fix \<P> P
-        assume "\<P> \<in> carrier (union_ring C')" and P: "P \<in> carrier (poly_ring R)" "\<not> index_free \<P> P"
-        from \<open>\<P> \<in> carrier (union_ring C')\<close> obtain T where T: "T \<in> C'" "\<P> \<in> carrier T"
-          using exists_superset_carrier[of C' "{ \<P> }"] core_chain by auto
-        hence "\<X>\<^bsub>P\<^esub> \<in> carrier T" and "(ring.eval T) (\<sigma> P) \<X>\<^bsub>P\<^esub> = \<zero>\<^bsub>T\<^esub>"
-          and field: "field T" and hom: "indexed_const \<in> ring_hom R T"
-          using P C'(1) unfolding extensions_def by auto
-        with \<open>T \<in> C'\<close> show "\<X>\<^bsub>P\<^esub> \<in> carrier (union_ring C')"
-          unfolding union_ring_carrier by auto
-        have "set P \<subseteq> carrier R"
-          using P(1) unfolding sym[OF univ_poly_carrier] polynomial_def by auto
-        hence "set (\<sigma> P) \<subseteq> carrier T"
-          using ring_hom_memE(1)[OF hom] unfolding \<sigma>_def by (induct P) (auto)
-        with \<open>\<X>\<^bsub>P\<^esub> \<in> carrier T\<close> and \<open>(ring.eval T) (\<sigma> P) \<X>\<^bsub>P\<^esub> = \<zero>\<^bsub>T\<^esub>\<close>
-        show "(ring.eval (union_ring C')) (\<sigma> P) \<X>\<^bsub>P\<^esub> = \<zero>\<^bsub>union_ring C'\<^esub>"
-          using iso_incl_imp_same_eval[OF field.is_ring[OF field] Union.is_ring
-                union_ring_is_upper_bound[OF core_chain T(1)]] same_one_same_zero(2)[OF core_chain T(1)]
-          by auto
-      qed
-      moreover have "R \<lesssim> law_restrict (union_ring C')" if "R \<in> C" for R
-        using that union_ring_is_upper_bound[OF core_chain] iso_incl_hom unfolding C' by auto
-      ultimately show ?thesis
-        by blast
+    have "union_ring C' \<in> extensions"
+    proof (auto simp add: extensions_def)
+      show "field (union_ring C')"
+        using Union.field_axioms .
+    next
+      from \<open>S \<in> C'\<close> have "indexed_const \<in> ring_hom R S"
+        using C'(1) unfolding extensions_def by auto
+      thus "indexed_const \<in> ring_hom R (union_ring C')"
+        using ring_hom_trans[of _ R S id] union_ring_is_upper_bound[OF core_chain S]
+        unfolding iso_incl.simps by auto
+    next
+      show "a \<in> carrier (union_ring C') \<Longrightarrow> carrier_coeff a" for a
+        using C'(1) unfolding union_ring_carrier extensions_def by auto
+    next
+      fix \<P> P i
+      assume "\<P> \<in> carrier (union_ring C')"
+        and P: "P \<in> carrier (poly_ring R)"
+        and not_index_free: "\<not> index_free \<P> (P, i)"
+      from \<open>\<P> \<in> carrier (union_ring C')\<close> obtain T where T: "T \<in> C'" "\<P> \<in> carrier T"
+        using exists_superset_carrier[of C' "{ \<P> }"] core_chain by auto
+      hence "\<X>\<^bsub>(P, i)\<^esub> \<in> carrier T" and "(ring.eval T) (\<sigma> P) \<X>\<^bsub>(P, i)\<^esub> = \<zero>\<^bsub>T\<^esub>"
+        and field: "field T" and hom: "indexed_const \<in> ring_hom R T"
+        using P not_index_free C'(1) unfolding extensions_def by auto
+      with \<open>T \<in> C'\<close> show "\<X>\<^bsub>(P, i)\<^esub> \<in> carrier (union_ring C')"
+        unfolding union_ring_carrier by auto
+      have "set P \<subseteq> carrier R"
+        using P unfolding sym[OF univ_poly_carrier] polynomial_def by auto
+      hence "set (\<sigma> P) \<subseteq> carrier T"
+        using ring_hom_memE(1)[OF hom] unfolding \<sigma>_def by (induct P) (auto)
+      with \<open>\<X>\<^bsub>(P, i)\<^esub> \<in> carrier T\<close> and \<open>(ring.eval T) (\<sigma> P) \<X>\<^bsub>(P, i)\<^esub> = \<zero>\<^bsub>T\<^esub>\<close>
+      show "(ring.eval (union_ring C')) (\<sigma> P) \<X>\<^bsub>(P, i)\<^esub> = \<zero>\<^bsub>union_ring C'\<^esub>"
+        using iso_incl_imp_same_eval[OF field.is_ring[OF field] Union.is_ring
+              union_ring_is_upper_bound[OF core_chain T(1)]] same_one_same_zero(2)[OF core_chain T(1)]
+        by auto
qed
+    moreover have "R \<lesssim> law_restrict (union_ring C')" if "R \<in> C" for R
+      using that union_ring_is_upper_bound[OF core_chain] iso_incl_hom unfolding C' by auto
+    ultimately show ?thesis
+      by blast
qed
qed

@@ -461,11 +468,11 @@
hence "set (map h p) \<subseteq> carrier S"
by (induct p) (auto)
moreover have "h a = \<zero>\<^bsub>S\<^esub> \<Longrightarrow> a = \<zero>\<^bsub>R\<^esub>" if "a \<in> carrier R" for a
-    using non_trivial_field_hom_is_inj[OF assms(1-3)] that unfolding inj_on_def by simp
+    using non_trivial_field_hom_is_inj[OF assms(1-3)] that unfolding inj_on_def by simp
with \<open>set p \<subseteq> carrier R\<close> have "lead_coeff (map h p) \<noteq> \<zero>\<^bsub>S\<^esub>" if "p \<noteq> []"
using lc[OF that] that by (cases p) (auto)
ultimately show ?thesis
-    unfolding sym[OF univ_poly_carrier] polynomial_def by auto
+    unfolding sym[OF univ_poly_carrier] polynomial_def by auto
qed

lemma (in ring_hom_ring) subfield_polynomial_hom:
@@ -485,44 +492,1252 @@
using S.univ_poly_consistent[OF subfieldE(1)[OF img_is_subfield(2)[OF assms]]] by simp
qed

+
+(* MOVE ========== *)
+subsection \<open>Roots and Multiplicity\<close>
+
+definition (in ring) is_root :: "'a list \<Rightarrow> 'a \<Rightarrow> bool"
+  where "is_root p x \<longleftrightarrow> (x \<in> carrier R \<and> eval p x = \<zero> \<and> p \<noteq> [])"
+
+definition (in ring) alg_mult :: "'a list \<Rightarrow> 'a \<Rightarrow> nat"
+  where "alg_mult p x =
+           (if p = [] then 0 else
+             (if x \<in> carrier R then Greatest (\<lambda> n. ([ \<one>, \<ominus> x ] [^]\<^bsub>poly_ring R\<^esub> n) pdivides p) else 0))"
+
+definition (in ring) roots :: "'a list \<Rightarrow> 'a multiset"
+  where "roots p = Abs_multiset (alg_mult p)"
+
+definition (in ring) roots_on :: "'a set \<Rightarrow> 'a list \<Rightarrow> 'a multiset"
+  where "roots_on K p = roots p \<inter># mset_set K"
+
+definition (in ring) splitted :: "'a list \<Rightarrow> bool"
+  where "splitted p \<longleftrightarrow> size (roots p) = degree p"
+
+definition (in ring) splitted_on :: "'a set \<Rightarrow> 'a list \<Rightarrow> bool"
+  where "splitted_on K p \<longleftrightarrow> size (roots_on K p) = degree p"
+
+lemma (in domain) polynomial_pow_not_zero:
+  assumes "p \<in> carrier (poly_ring R)" and "p \<noteq> []"
+  shows "p [^]\<^bsub>poly_ring R\<^esub> (n::nat) \<noteq> []"
+proof -
+  interpret UP: domain "poly_ring R"
+    using univ_poly_is_domain[OF carrier_is_subring] .
+
+  from assms UP.integral show ?thesis
+    unfolding sym[OF univ_poly_zero[of R "carrier R"]]
+    by (induction n, auto)
+qed
+
+lemma (in domain) subring_polynomial_pow_not_zero:
+  assumes "subring K R" and "p \<in> carrier (K[X])" and "p \<noteq> []"
+  shows "p [^]\<^bsub>K[X]\<^esub> (n::nat) \<noteq> []"
+  using domain.polynomial_pow_not_zero[OF subring_is_domain, of K p n] assms
+  unfolding univ_poly_consistent[OF assms(1)] by simp
+
+lemma (in domain) polynomial_pow_degree:
+  assumes "p \<in> carrier (poly_ring R)"
+  shows "degree (p [^]\<^bsub>poly_ring R\<^esub> n) = n * degree p"
+proof -
+  interpret UP: domain "poly_ring R"
+    using univ_poly_is_domain[OF carrier_is_subring] .
+
+  show ?thesis
+  proof (induction n)
+    case 0 thus ?case
+      using UP.nat_pow_0 unfolding univ_poly_one by auto
+  next
+    let ?ppow = "\<lambda>n. p [^]\<^bsub>poly_ring R\<^esub> n"
+    case (Suc n) thus ?case
+    proof (cases "p = []")
+      case True thus ?thesis
+        using univ_poly_zero[of R "carrier R"] UP.r_null assms by auto
+    next
+      case False
+      hence "?ppow n \<in> carrier (poly_ring R)" and "?ppow n \<noteq> []" and "p \<noteq> []"
+        using polynomial_pow_not_zero[of p n] assms by (auto simp add: univ_poly_one)
+      thus ?thesis
+        using poly_mult_degree_eq[OF carrier_is_subring, of "?ppow n" p] Suc assms
+        unfolding univ_poly_carrier univ_poly_zero
+    qed
+  qed
+qed
+
+lemma (in domain) polynomial_pow_division:
+  assumes "p \<in> carrier (poly_ring R)" and "(n::nat) \<le> m"
+  shows "(p [^]\<^bsub>poly_ring R\<^esub> n) pdivides (p [^]\<^bsub>poly_ring R\<^esub> m)"
+proof -
+  interpret UP: domain "poly_ring R"
+    using univ_poly_is_domain[OF carrier_is_subring] .
+
+  let ?ppow = "\<lambda>n. p [^]\<^bsub>poly_ring R\<^esub> n"
+
+  have "?ppow n \<otimes>\<^bsub>poly_ring R\<^esub> ?ppow k = ?ppow (n + k)" for k
+    using assms(1) by (simp add: UP.nat_pow_mult)
+  thus ?thesis
+    using dividesI[of "?ppow (m - n)" "poly_ring R" "?ppow m" "?ppow n"] assms
+    unfolding pdivides_def by auto
+qed
+
+lemma (in domain) degree_zero_imp_not_is_root:
+  assumes "p \<in> carrier (poly_ring R)" and "degree p = 0" shows "\<not> is_root p x"
+proof (cases "p = []", simp add: is_root_def)
+  case False with \<open>degree p = 0\<close> have "length p = Suc 0"
+    using le_SucE by fastforce
+  then obtain a where "p = [ a ]" and "a \<in> carrier R" and "a \<noteq> \<zero>"
+    using assms unfolding sym[OF univ_poly_carrier] polynomial_def
+    by (metis False length_0_conv length_Suc_conv list.sel(1) list.set_sel(1) subset_code(1))
+  thus ?thesis
+    unfolding is_root_def by auto
+qed
+
+lemma (in domain) is_root_imp_pdivides:
+  assumes "p \<in> carrier (poly_ring R)"
+  shows "is_root p x \<Longrightarrow> [ \<one>, \<ominus> x ] pdivides p"
+proof -
+  let ?b = "[ \<one> , \<ominus> x ]"
+
+  interpret UP: domain "poly_ring R"
+    using univ_poly_is_domain[OF carrier_is_subring] .
+
+  assume "is_root p x" hence x: "x \<in> carrier R" and is_root: "eval p x = \<zero>"
+    unfolding is_root_def by auto
+  hence b: "?b \<in> carrier (poly_ring R)"
+    unfolding sym[OF univ_poly_carrier] polynomial_def by auto
+  then obtain q r where q: "q \<in> carrier (poly_ring R)" and r: "r \<in> carrier (poly_ring R)"
+    and long_divides: "p = (?b \<otimes>\<^bsub>poly_ring R\<^esub> q) \<oplus>\<^bsub>poly_ring R\<^esub> r" "r = [] \<or> degree r < degree ?b"
+    using long_division_theorem[OF carrier_is_subring, of p ?b] assms by (auto simp add: univ_poly_carrier)
+
+  show ?thesis
+  proof (cases "r = []")
+    case True then have "r = \<zero>\<^bsub>poly_ring R\<^esub>"
+      unfolding univ_poly_zero[of R "carrier R"] .
+    thus ?thesis
+      using long_divides(1) q r b dividesI[OF q, of p ?b] by (simp add: pdivides_def)
+  next
+    case False then have "length r = Suc 0"
+      using long_divides(2) le_SucE by fastforce
+    then obtain a where "r = [ a ]" and a: "a \<in> carrier R" and "a \<noteq> \<zero>"
+      using r unfolding sym[OF univ_poly_carrier] polynomial_def
+      by (metis False length_0_conv length_Suc_conv list.sel(1) list.set_sel(1) subset_code(1))
+
+    have "eval p x = ((eval ?b x) \<otimes> (eval q x)) \<oplus> (eval r x)"
+      using long_divides(1) ring_hom_memE[OF eval_is_hom[OF carrier_is_subring x]] by (simp add: b q r)
+    also have " ... = eval r x"
+      using ring_hom_memE[OF eval_is_hom[OF carrier_is_subring x]] x b q r by (auto, algebra)
+    finally have "a = \<zero>"
+      using a unfolding \<open>r = [ a ]\<close> is_root by simp
+    with \<open>a \<noteq> \<zero>\<close> have False .. thus ?thesis ..
+  qed
+qed
+
+lemma (in domain) pdivides_imp_is_root:
+  assumes "p \<noteq> []" and "x \<in> carrier R"
+  shows "[ \<one>, \<ominus> x ] pdivides p \<Longrightarrow> is_root p x"
+proof -
+  assume "[ \<one>, \<ominus> x ] pdivides p"
+  then obtain q where q: "q \<in> carrier (poly_ring R)" and pdiv: "p = [ \<one>, \<ominus> x ] \<otimes>\<^bsub>poly_ring R\<^esub> q"
+    unfolding pdivides_def by auto
+  moreover have "[ \<one>, \<ominus> x ] \<in> carrier (poly_ring R)"
+    using assms(2) unfolding sym[OF univ_poly_carrier] polynomial_def by simp
+  ultimately have "eval p x = \<zero>"
+    using ring_hom_memE[OF eval_is_hom[OF carrier_is_subring, of x]] assms(2) by (auto, algebra)
+  with \<open>p \<noteq> []\<close> and \<open>x \<in> carrier R\<close> show "is_root p x"
+    unfolding is_root_def by simp
+qed
+
+(* MOVE TO Polynomial_Dvisibility.thy ================== *)
+lemma (in domain) associated_polynomials_imp_same_length: (* stronger than "imp_same_degree" *)
+  assumes "subring K R" and "p \<in> carrier (K[X])" and "q \<in> carrier (K[X])"
+  shows "p \<sim>\<^bsub>K[X]\<^esub> q \<Longrightarrow> length p = length q"
+proof -
+  { fix p q
+    assume p: "p \<in> carrier (K[X])" and q: "q \<in> carrier (K[X])" and "p \<sim>\<^bsub>K[X]\<^esub> q"
+    have "length p \<le> length q"
+    proof (cases "q = []")
+      case True with \<open>p \<sim>\<^bsub>K[X]\<^esub> q\<close> have "p = []"
+        unfolding associated_def True factor_def univ_poly_def by auto
+      thus ?thesis
+        using True by simp
+    next
+      case False
+      from \<open>p \<sim>\<^bsub>K[X]\<^esub> q\<close> have "p divides\<^bsub>K [X]\<^esub> q"
+        unfolding associated_def by simp
+      hence "p divides\<^bsub>poly_ring R\<^esub> q"
+        using carrier_polynomial[OF assms(1)]
+        unfolding factor_def univ_poly_carrier univ_poly_mult by auto
+      with \<open>q \<noteq> []\<close> have "degree p \<le> degree q"
+        using pdivides_imp_degree_le[OF assms(1) p q] unfolding pdivides_def by simp
+      with \<open>q \<noteq> []\<close> show ?thesis
+        by (cases "p = []", auto simp add: Suc_leI le_diff_iff)
+    qed
+  } note aux_lemma = this
+
+  interpret UP: domain "K[X]"
+    using univ_poly_is_domain[OF assms(1)] .
+
+  assume "p \<sim>\<^bsub>K[X]\<^esub> q" thus ?thesis
+    using aux_lemma[OF assms(2-3)] aux_lemma[OF assms(3,2) UP.associated_sym] by simp
+qed
+(* ================================================= *)
+
+lemma (in domain) associated_polynomials_imp_same_is_root:
+  assumes "p \<in> carrier (poly_ring R)" and "q \<in> carrier (poly_ring R)" and "p \<sim>\<^bsub>poly_ring R\<^esub> q"
+  shows "is_root p x \<longleftrightarrow> is_root q x"
+proof (cases "p = []")
+  case True with \<open>p \<sim>\<^bsub>poly_ring R\<^esub> q\<close> have "q = []"
+    unfolding associated_def True factor_def univ_poly_def by auto
+  thus ?thesis
+    using True unfolding is_root_def by simp
+next
+  case False
+  interpret UP: domain "poly_ring R"
+    using univ_poly_is_domain[OF carrier_is_subring] .
+
+  { fix p q
+    assume p: "p \<in> carrier (poly_ring R)" and q: "q \<in> carrier (poly_ring R)" and pq: "p \<sim>\<^bsub>poly_ring R\<^esub> q"
+    have "is_root p x \<Longrightarrow> is_root q x"
+    proof -
+      assume is_root: "is_root p x"
+      then have "[ \<one>, \<ominus> x ] pdivides p" and "p \<noteq> []" and "x \<in> carrier R"
+        using is_root_imp_pdivides[OF p] unfolding is_root_def by auto
+      moreover have "[ \<one>, \<ominus> x ] \<in> carrier (poly_ring R)"
+        using is_root unfolding is_root_def sym[OF univ_poly_carrier] polynomial_def by simp
+      ultimately have "[ \<one>, \<ominus> x ] pdivides q"
+        using UP.divides_cong_r[OF _ pq ] unfolding pdivides_def by simp
+      with \<open>p \<noteq> []\<close> and \<open>x \<in> carrier R\<close> show ?thesis
+        using associated_polynomials_imp_same_length[OF carrier_is_subring p q pq]
+              pdivides_imp_is_root[of q x]
+        by fastforce
+    qed
+  }
+
+  then show ?thesis
+    using assms UP.associated_sym[OF assms(3)] by blast
+qed
+
+lemma (in ring) monic_degree_one_root_condition:
+  assumes "a \<in> carrier R" shows "is_root [ \<one>, \<ominus> a ] b \<longleftrightarrow> a = b"
+  using assms minus_equality r_neg[OF assms] unfolding is_root_def by (auto, fastforce)
+
+lemma (in field) degree_one_root_condition:
+  assumes "p \<in> carrier (poly_ring R)" and "degree p = 1"
+  shows "is_root p x \<longleftrightarrow> x = \<ominus> (inv (lead_coeff p) \<otimes> (const_term p))"
+proof -
+  interpret UP: domain "poly_ring R"
+    using univ_poly_is_domain[OF carrier_is_subring] .
+
+  from \<open>degree p = 1\<close> have "length p = Suc (Suc 0)"
+    by simp
+  then obtain a b where p: "p = [ a, b ]"
+    by (metis length_0_conv length_Cons list.exhaust nat.inject)
+  hence a: "a \<in> carrier R" "a \<noteq> \<zero>" and b: "b \<in> carrier R"
+    using assms(1) unfolding sym[OF univ_poly_carrier] polynomial_def by auto
+  hence inv_a: "inv a \<in> carrier R" "(inv a) \<otimes> a = \<one>"
+    using subfield_m_inv[OF carrier_is_subfield, of a] by auto
+  hence in_carrier: "[ \<one>, (inv a) \<otimes> b ] \<in> carrier (poly_ring R)"
+    using b unfolding sym[OF univ_poly_carrier] polynomial_def by auto
+
+  have "p \<sim>\<^bsub>poly_ring R\<^esub> [ \<one>, (inv a) \<otimes> b ]"
+  proof (rule UP.associatedI2'[OF _ _ in_carrier, of _ "[ a ]"])
+    have "p = [ a ] \<otimes>\<^bsub>poly_ring R\<^esub> [ \<one>, inv a \<otimes> b ]"
+      using a inv_a b m_assoc[of a "inv a" b] unfolding p univ_poly_mult by (auto, algebra)
+    also have " ... = [ \<one>, inv a \<otimes> b ] \<otimes>\<^bsub>poly_ring R\<^esub> [ a ]"
+      using UP.m_comm[OF in_carrier, of "[ a ]"] a
+      by (auto simp add: sym[OF univ_poly_carrier] polynomial_def)
+    finally show "p = [ \<one>, inv a \<otimes> b ] \<otimes>\<^bsub>poly_ring R\<^esub> [ a ]" .
+  next
+    from \<open>a \<in> carrier R\<close> and \<open>a \<noteq> \<zero>\<close> show "[ a ] \<in> Units (poly_ring R)"
+      unfolding univ_poly_units[OF carrier_is_subfield] by simp
+  qed
+
+  moreover have "(inv a) \<otimes> b = \<ominus> (\<ominus> (inv (lead_coeff p) \<otimes> (const_term p)))"
+    and "inv (lead_coeff p) \<otimes> (const_term p) \<in> carrier R"
+    using inv_a a b unfolding p const_term_def by auto
+
+  ultimately show ?thesis
+    using associated_polynomials_imp_same_is_root[OF assms(1) in_carrier]
+          monic_degree_one_root_condition
+qed
+
+lemma (in domain) is_root_imp_is_root_poly_mult:
+  assumes "p \<in> carrier (poly_ring R)" and "q \<in> carrier (poly_ring R)" and "q \<noteq> []"
+  shows "is_root p x \<Longrightarrow> is_root (p \<otimes>\<^bsub>poly_ring R\<^esub> q) x"
+proof -
+  interpret UP: domain "poly_ring R"
+    using univ_poly_is_domain[OF carrier_is_subring] .
+
+  assume "is_root p x" then have x: "x \<in> carrier R" and eval: "eval p x = \<zero>" and not_zero: "p \<noteq> []"
+    unfolding is_root_def by simp+
+  hence "p \<otimes>\<^bsub>poly_ring R\<^esub> q \<noteq> []"
+    using assms UP.integral unfolding sym[OF univ_poly_zero[of R "carrier R"]] by blast
+  moreover have "eval (p \<otimes>\<^bsub>poly_ring R\<^esub> q) x = \<zero>"
+    using assms eval ring_hom_memE[OF eval_is_hom[OF carrier_is_subring x]] by auto
+  ultimately show ?thesis
+    using x unfolding is_root_def by simp
+qed
+
+lemma (in domain) is_root_poly_mult_imp_is_root:
+  assumes "p \<in> carrier (poly_ring R)" and "q \<in> carrier (poly_ring R)"
+  shows "is_root (p \<otimes>\<^bsub>poly_ring R\<^esub> q) x \<Longrightarrow> (is_root p x) \<or> (is_root q x)"
+proof -
+  interpret UP: domain "poly_ring R"
+    using univ_poly_is_domain[OF carrier_is_subring] .
+
+  assume is_root: "is_root (p \<otimes>\<^bsub>poly_ring R\<^esub> q) x"
+  hence "p \<noteq> []" and "q \<noteq> []"
+    unfolding is_root_def sym[OF univ_poly_zero[of R "carrier R"]]
+    using UP.l_null[OF assms(2)] UP.r_null[OF assms(1)] by blast+
+  moreover have x: "x \<in> carrier R" and "eval (p \<otimes>\<^bsub>poly_ring R\<^esub> q) x = \<zero>"
+    using is_root unfolding is_root_def by simp+
+  hence "eval p x = \<zero> \<or> eval q x = \<zero>"
+    using ring_hom_memE[OF eval_is_hom[OF carrier_is_subring], of x] assms integral by auto
+  ultimately show "(is_root p x) \<or> (is_root q x)"
+    using x unfolding is_root_def by auto
+qed
+
+(*
+lemma (in domain)
+  assumes "subfield K R" and "p \<in> carrier (K[X])" and "degree p = 1"
+  shows "pprime K p"
+proof (rule pprimeI[OF assms(1-2)])
+  from \<open>degree p = 1\<close> show "p \<noteq> []" and "p \<notin> Units (K [X])"
+    unfolding univ_poly_units[OF assms(1)] by auto
+next
+  fix q r
+  assume "q \<in> carrier (K[X])" and "r \<in> carrier (K[X])"
+    and pdiv: "p pdivides q \<otimes>\<^bsub>K [X]\<^esub> r"
+  hence "q \<in> carrier (poly_ring R)" and "r \<in> carrier (poly_ring R)"
+    using carrier_polynomial_shell[OF subfieldE(1)[OF assms(1)]] by auto
+  moreover obtain b where b: "b \<in>"
+qed
+*)
+
+lemma (in domain) finite_number_of_roots:
+  assumes "p \<in> carrier (poly_ring R)" shows "finite { x. is_root p x }"
+  using assms
+proof (induction "degree p" arbitrary: p)
+  case 0 thus ?case
+next
+  case (Suc n) show ?case
+  proof (cases "{ x. is_root p x } = {}")
+    case True thus ?thesis
+  next
+    interpret UP: domain "poly_ring R"
+      using univ_poly_is_domain[OF carrier_is_subring] .
+
+    case False
+    then obtain a where is_root: "is_root p a"
+      by blast
+    hence a: "a \<in> carrier R" and eval: "eval p a = \<zero>" and p_not_zero: "p \<noteq> []"
+      unfolding is_root_def by auto
+    hence in_carrier: "[ \<one>, \<ominus> a ] \<in> carrier (poly_ring R)"
+      unfolding sym[OF univ_poly_carrier] polynomial_def by auto
+
+    obtain q where q: "q \<in> carrier (poly_ring R)" and p: "p = [ \<one>, \<ominus> a ] \<otimes>\<^bsub>poly_ring R\<^esub> q"
+      using is_root_imp_pdivides[OF Suc(3) is_root] unfolding pdivides_def by auto
+    with \<open>p \<noteq> []\<close> have q_not_zero: "q \<noteq> []"
+      using UP.r_null UP.integral in_carrier unfolding sym[OF univ_poly_zero[of R "carrier R"]]
+      by metis
+    hence "degree q = n"
+      using poly_mult_degree_eq[OF carrier_is_subring, of "[ \<one>, \<ominus> a ]" q]
+            in_carrier q p_not_zero p Suc(2)
+      unfolding univ_poly_carrier
+      by (metis One_nat_def Suc_eq_plus1 diff_Suc_1 list.distinct(1)
+                list.size(3-4) plus_1_eq_Suc univ_poly_mult)
+    hence "finite { x. is_root q x }"
+      using Suc(1)[OF _ q] by simp
+
+    moreover have "{ x. is_root p x } \<subseteq> insert a { x. is_root q x }"
+      using is_root_poly_mult_imp_is_root[OF in_carrier q]
+            monic_degree_one_root_condition[OF a]
+      unfolding p by auto
+
+    ultimately show ?thesis
+      using finite_subset by auto
+  qed
+qed
+
+lemma (in domain) alg_multE:
+  assumes "x \<in> carrier R" and "p \<in> carrier (poly_ring R)" and "p \<noteq> []"
+  shows "([ \<one>, \<ominus> x ] [^]\<^bsub>poly_ring R\<^esub> (alg_mult p x)) pdivides p"
+    and "\<And>n. ([ \<one>, \<ominus> x ] [^]\<^bsub>poly_ring R\<^esub> n) pdivides p \<Longrightarrow> n \<le> alg_mult p x"
+proof -
+  interpret UP: domain "poly_ring R"
+    using univ_poly_is_domain[OF carrier_is_subring] .
+
+  let ?ppow = "\<lambda>n :: nat. ([ \<one>, \<ominus> x ] [^]\<^bsub>poly_ring R\<^esub> n)"
+
+  define S :: "nat set" where "S = { n. ?ppow n pdivides p }"
+  have "?ppow 0 = \<one>\<^bsub>poly_ring R\<^esub>"
+    using UP.nat_pow_0 by simp
+  hence "0 \<in> S"
+    using UP.one_divides[OF assms(2)] unfolding S_def pdivides_def by simp
+  hence "S \<noteq> {}"
+    by auto
+
+  moreover have "n \<le> degree p" if "n \<in> S" for n :: nat
+  proof -
+    have "[ \<one>, \<ominus> x ] \<in> carrier (poly_ring R)"
+      using assms unfolding sym[OF univ_poly_carrier] polynomial_def by auto
+    hence "?ppow n \<in> carrier (poly_ring R)"
+      using assms unfolding univ_poly_zero by auto
+    with \<open>n \<in> S\<close> have "degree (?ppow n) \<le> degree p"
+      using pdivides_imp_degree_le[OF carrier_is_subring _ assms(2-3), of "?ppow n"] by (simp add: S_def)
+    with \<open>[ \<one>, \<ominus> x ] \<in> carrier (poly_ring R)\<close> show ?thesis
+      using polynomial_pow_degree by simp
+  qed
+  hence "finite S"
+    using finite_nat_set_iff_bounded_le by blast
+
+  ultimately have MaxS: "\<And>n. n \<in> S \<Longrightarrow> n \<le> Max S" "Max S \<in> S"
+    using Max_ge[of S] Max_in[of S] by auto
+  with \<open>x \<in> carrier R\<close> have "alg_mult p x = Max S"
+    using Greatest_equality[of "\<lambda>n. ?ppow n pdivides p" "Max S"] assms(3)
+    unfolding S_def alg_mult_def by auto
+  thus "([ \<one>, \<ominus> x ] [^]\<^bsub>poly_ring R\<^esub> (alg_mult p x)) pdivides p"
+   and "\<And>n. ([ \<one>, \<ominus> x ] [^]\<^bsub>poly_ring R\<^esub> n) pdivides p \<Longrightarrow> n \<le> alg_mult p x"
+    using MaxS unfolding S_def by auto
+qed
+
+lemma (in domain) le_alg_mult_imp_pdivides:
+  assumes "x \<in> carrier R" and "p \<in> carrier (poly_ring R)"
+  shows "n \<le> alg_mult p x \<Longrightarrow> ([ \<one>, \<ominus> x ] [^]\<^bsub>poly_ring R\<^esub> n) pdivides p"
+proof -
+  interpret UP: domain "poly_ring R"
+    using univ_poly_is_domain[OF carrier_is_subring] .
+
+  assume le_alg_mult: "n \<le> alg_mult p x"
+  have in_carrier: "[ \<one>, \<ominus> x ] \<in> carrier (poly_ring R)"
+    using assms(1) unfolding sym[OF univ_poly_carrier] polynomial_def by auto
+  hence ppow_pdivides:
+    "([ \<one>, \<ominus> x ] [^]\<^bsub>poly_ring R\<^esub> n) pdivides
+     ([ \<one>, \<ominus> x ] [^]\<^bsub>poly_ring R\<^esub> (alg_mult p x))"
+    using polynomial_pow_division[OF _ le_alg_mult] by simp
+
+  show ?thesis
+  proof (cases "p = []")
+    case True thus ?thesis
+      using in_carrier pdivides_zero[OF carrier_is_subring] by auto
+  next
+    case False thus ?thesis
+      using ppow_pdivides UP.divides_trans UP.nat_pow_closed alg_multE(1)[OF assms] in_carrier
+      unfolding pdivides_def by meson
+  qed
+qed
+
+lemma (in domain) alg_mult_gt_zero_iff_is_root:
+  assumes "p \<in> carrier (poly_ring R)" shows "alg_mult p x > 0 \<longleftrightarrow> is_root p x"
+proof -
+  interpret UP: domain "poly_ring R"
+    using univ_poly_is_domain[OF carrier_is_subring] .
+  show ?thesis
+  proof
+    assume is_root: "is_root p x" hence x: "x \<in> carrier R" and not_zero: "p \<noteq> []"
+      unfolding is_root_def by auto
+    have "[\<one>, \<ominus> x] [^]\<^bsub>poly_ring R\<^esub> (Suc 0) = [\<one>, \<ominus> x]"
+      using x unfolding univ_poly_def by auto
+    thus "alg_mult p x > 0"
+      using is_root_imp_pdivides[OF _ is_root] alg_multE(2)[OF x, of p "Suc 0"] not_zero assms by auto
+  next
+    assume gt_zero: "alg_mult p x > 0"
+    hence x: "x \<in> carrier R" and not_zero: "p \<noteq> []"
+      unfolding alg_mult_def by (cases "p = []", auto, cases "x \<in> carrier R", auto)
+    hence in_carrier: "[ \<one>, \<ominus> x ] \<in> carrier (poly_ring R)"
+      unfolding sym[OF univ_poly_carrier] polynomial_def by auto
+    with \<open>x \<in> carrier R\<close> have "[ \<one>, \<ominus> x ] pdivides p" and "eval [ \<one>, \<ominus> x ] x = \<zero>"
+      using le_alg_mult_imp_pdivides[of x p "1::nat"] gt_zero assms by (auto, algebra)
+    thus "is_root p x"
+      using pdivides_imp_root_sharing[OF in_carrier] not_zero x by (simp add: is_root_def)
+  qed
+qed
+
+lemma (in domain) alg_mult_in_multiset:
+  assumes "p \<in> carrier (poly_ring R)" shows "alg_mult p \<in> multiset"
+  using assms alg_mult_gt_zero_iff_is_root finite_number_of_roots unfolding multiset_def by auto
+
+lemma (in domain) alg_mult_eq_count_roots:
+  assumes "p \<in> carrier (poly_ring R)" shows "alg_mult p = count (roots p)"
+  using alg_mult_in_multiset[OF assms] by (simp add: roots_def)
+
+lemma (in domain) roots_mem_iff_is_root:
+  assumes "p \<in> carrier (poly_ring R)" shows "x \<in># roots p \<longleftrightarrow> is_root p x"
+  using alg_mult_eq_count_roots[OF assms] count_greater_zero_iff
+  unfolding roots_def sym[OF alg_mult_gt_zero_iff_is_root[OF assms]] by metis
+
+lemma (in domain) degree_zero_imp_empty_roots:
+  assumes "p \<in> carrier (poly_ring R)" and "degree p = 0" shows "roots p = {#}"
+proof -
+  have "alg_mult p = (\<lambda>x. 0)"
+  proof (cases "p = []")
+    case True thus ?thesis
+      using assms unfolding alg_mult_def by auto
+  next
+    case False hence "length p = Suc 0"
+      using assms(2) by (simp add: le_Suc_eq)
+    then obtain a where "a \<in> carrier R" and "a \<noteq> \<zero>" and p: "p = [ a ]"
+      using assms(1) unfolding sym[OF univ_poly_carrier] polynomial_def
+      by (metis Suc_length_conv hd_in_set length_0_conv list.sel(1) subset_code(1))
+    show ?thesis
+    proof (rule ccontr)
+      assume "alg_mult p \<noteq> (\<lambda>x. 0)"
+      then obtain x where "alg_mult p x > 0"
+        by auto
+      with \<open>p \<noteq> []\<close> have "eval p x = \<zero>"
+        using alg_mult_gt_zero_iff_is_root[OF assms(1), of x] unfolding is_root_def by simp
+      with \<open>a \<in> carrier R\<close> have "a = \<zero>"
+        unfolding p by auto
+      with \<open>a \<noteq> \<zero>\<close> show False ..
+    qed
+  qed
+  thus ?thesis
+    by (simp add: roots_def zero_multiset.abs_eq)
+qed
+
+lemma (in domain) degree_zero_imp_splitted:
+  assumes "p \<in> carrier (poly_ring R)" and "degree p = 0" shows "splitted p"
+  unfolding splitted_def degree_zero_imp_empty_roots[OF assms] assms(2) by simp
+
+lemma (in domain) roots_inclI':
+  assumes "p \<in> carrier (poly_ring R)" and "\<And>a. \<lbrakk> a \<in> carrier R; p \<noteq> [] \<rbrakk> \<Longrightarrow> alg_mult p a \<le> count m a"
+  shows "roots p \<subseteq># m"
+proof (intro mset_subset_eqI)
+  fix a show "count (roots p) a \<le> count m a"
+    using assms unfolding sym[OF alg_mult_eq_count_roots[OF assms(1)]] alg_mult_def
+    by (cases "p = []", simp, cases "a \<in> carrier R", auto)
+qed
+
+lemma (in domain) roots_inclI:
+  assumes "p \<in> carrier (poly_ring R)" and "q \<in> carrier (poly_ring R)" "q \<noteq> []"
+    and "\<And>a. \<lbrakk> a \<in> carrier R; p \<noteq> [] \<rbrakk> \<Longrightarrow> ([ \<one>, \<ominus> a ] [^]\<^bsub>poly_ring R\<^esub> (alg_mult p a)) pdivides q"
+  shows "roots p \<subseteq># roots q"
+  using roots_inclI'[OF assms(1), of "roots q"] assms alg_multE(2)[OF _ assms(2-3)]
+  unfolding sym[OF alg_mult_eq_count_roots[OF assms(2)]] by auto
+
+lemma (in domain) pdivides_imp_roots_incl:
+  assumes "p \<in> carrier (poly_ring R)" and "q \<in> carrier (poly_ring R)" "q \<noteq> []"
+  shows "p pdivides q \<Longrightarrow> roots p \<subseteq># roots q"
+proof (rule roots_inclI[OF assms])
+  interpret UP: domain "poly_ring R"
+    using univ_poly_is_domain[OF carrier_is_subring] .
+
+  fix a assume "p pdivides q" and a: "a \<in> carrier R"
+  hence "[ \<one> , \<ominus> a ] \<in> carrier (poly_ring R)"
+    unfolding sym[OF univ_poly_carrier] polynomial_def by simp
+  with \<open>p pdivides q\<close> show "([\<one>, \<ominus> a] [^]\<^bsub>poly_ring R\<^esub> (alg_mult p a)) pdivides q"
+    using UP.divides_trans[of _p q] le_alg_mult_imp_pdivides[OF a assms(1)]
+    by (auto simp add: pdivides_def)
+qed
+
+lemma (in domain) associated_polynomials_imp_same_roots:
+  assumes "p \<in> carrier (poly_ring R)" and "q \<in> carrier (poly_ring R)" and "p \<sim>\<^bsub>poly_ring R\<^esub> q"
+  shows "roots p = roots q"
+  using assms pdivides_imp_roots_incl zero_pdivides
+  unfolding pdivides_def associated_def
+  by (metis subset_mset.eq_iff)
+
+(* MOVE to Polynomial_Divisibility.thy *)
+lemma (in comm_monoid_cancel) prime_pow_divides_iff:
+  assumes "p \<in> carrier G" "a \<in> carrier G" "b \<in> carrier G" and "prime G p" and "\<not> (p divides a)"
+  shows "(p [^] (n :: nat)) divides (a \<otimes> b) \<longleftrightarrow> (p [^] n) divides b"
+proof
+  assume "(p [^] n) divides b" thus "(p [^] n) divides (a \<otimes> b)"
+    using divides_prod_l[of "p [^] n" b a] assms by simp
+next
+  assume "(p [^] n) divides (a \<otimes> b)" thus "(p [^] n) divides b"
+  proof (induction n)
+    case 0 with \<open>b \<in> carrier G\<close> show ?case
+  next
+    case (Suc n)
+    hence "(p [^] n) divides (a \<otimes> b)" and "(p [^] n) divides b"
+      using assms(1) divides_prod_r by auto
+    with \<open>(p [^] (Suc n)) divides (a \<otimes> b)\<close> obtain c d
+      where c: "c \<in> carrier G" and "b = (p [^] n) \<otimes> c"
+        and d: "d \<in> carrier G" and "a \<otimes> b = (p [^] (Suc n)) \<otimes> d"
+      using assms by blast
+    hence "(p [^] n) \<otimes> (a \<otimes> c) = (p [^] n) \<otimes> (p \<otimes> d)"
+      using assms by (simp add: m_assoc m_lcomm)
+    hence "a \<otimes> c = p \<otimes> d"
+      using c d assms(1) assms(2) l_cancel by blast
+    with \<open>\<not> (p divides a)\<close> and \<open>prime G p\<close> have "p divides c"
+      by (metis assms(2) c d dividesI' prime_divides)
+    with \<open>b = (p [^] n) \<otimes> c\<close> show ?case
+      using assms(1) c by simp
+  qed
+qed
+
+(* MOVE to Polynomial_Divisibility.thy *)
+lemma (in domain) pirreducible_pow_pdivides_iff:
+  assumes "subfield K R" "p \<in> carrier (K[X])" "q \<in> carrier (K[X])" "r \<in> carrier (K[X])"
+    and "pirreducible K p" and "\<not> (p pdivides q)"
+  shows "(p [^]\<^bsub>K[X]\<^esub> (n :: nat)) pdivides (q \<otimes>\<^bsub>K[X]\<^esub> r) \<longleftrightarrow> (p [^]\<^bsub>K[X]\<^esub> n) pdivides r"
+proof -
+  interpret UP: principal_domain "K[X]"
+    using univ_poly_is_principal[OF assms(1)] .
+  show ?thesis
+  proof (cases "r = []")
+    case True with \<open>q \<in> carrier (K[X])\<close> have "q \<otimes>\<^bsub>K[X]\<^esub> r = []" and "r = []"
+      unfolding  sym[OF univ_poly_zero[of R K]] by auto
+    thus ?thesis
+      using pdivides_zero[OF subfieldE(1),of K] assms by auto
+  next
+    case False then have not_zero: "p \<noteq> []" "q \<noteq> []" "r \<noteq> []" "q \<otimes>\<^bsub>K[X]\<^esub> r \<noteq> []"
+      using subfieldE(1) pdivides_zero[OF _ assms(2)] assms(1-2,5-6) pirreducibleE(1)
+            UP.integral_iff[OF assms(3-4)] univ_poly_zero[of R K] by auto
+    from \<open>p \<noteq> []\<close>
+    have ppow: "p [^]\<^bsub>K[X]\<^esub> (n :: nat) \<noteq> []" "p [^]\<^bsub>K[X]\<^esub> (n :: nat) \<in> carrier (K[X])"
+      using subring_polynomial_pow_not_zero[OF subfieldE(1)] assms(1-2) by auto
+    have not_pdiv: "\<not> (p divides\<^bsub>mult_of (K[X])\<^esub> q)"
+      using assms(6) pdivides_iff_shell[OF assms(1-3)] unfolding pdivides_def by auto
+    have prime: "prime (mult_of (K[X])) p"
+      using assms(5) pprime_iff_pirreducible[OF assms(1-2)]
+      unfolding sym[OF UP.prime_eq_prime_mult[OF assms(2)]] ring_prime_def by simp
+    have "a pdivides b \<longleftrightarrow> a divides\<^bsub>mult_of (K[X])\<^esub> b"
+      if "a \<in> carrier (K[X])" "a \<noteq> \<zero>\<^bsub>K[X]\<^esub>" "b \<in> carrier (K[X])" "b \<noteq> \<zero>\<^bsub>K[X]\<^esub>" for a b
+      using that UP.divides_imp_divides_mult[of a b] divides_mult_imp_divides[of "K[X]" a b]
+      unfolding pdivides_iff_shell[OF assms(1) that(1,3)] by blast
+    thus ?thesis
+      using UP.mult_of.prime_pow_divides_iff[OF _ _ _ prime not_pdiv, of r] ppow not_zero assms(2-4)
+      unfolding nat_pow_mult_of carrier_mult_of mult_mult_of sym[OF univ_poly_zero[of R K]]
+      by (metis DiffI UP.m_closed singletonD)
+  qed
+qed
+
+(* MOVE to Polynomial_Divisibility.thy *)
+lemma (in domain) univ_poly_units':
+  assumes "subfield K R" shows "p \<in> Units (K[X]) \<longleftrightarrow> p \<in> carrier (K[X]) \<and> p \<noteq> [] \<and> degree p = 0"
+  unfolding univ_poly_units[OF assms] sym[OF univ_poly_carrier] polynomial_def
+  by (auto, metis hd_in_set le_0_eq le_Suc_eq length_0_conv length_Suc_conv list.sel(1) subsetD)
+
+(* MOVE to Polynomial_Divisibility.thy *)
+lemma (in domain) subring_degree_one_imp_pirreducible:
+  assumes "subring K R" and "a \<in> Units (R \<lparr> carrier := K \<rparr>)" and "b \<in> K"
+  shows "pirreducible K [ a, b ]"
+proof (rule pirreducibleI[OF assms(1)])
+  have "a \<in> K" and "a \<noteq> \<zero>"
+    using assms(2) subringE(1)[OF assms(1)] unfolding Units_def by auto
+  thus "[ a, b ] \<in> carrier (K[X])" and "[ a, b ] \<noteq> []" and "[ a, b ] \<notin> Units (K [X])"
+    using univ_poly_units_incl[OF assms(1)] assms(2-3)
+    unfolding sym[OF univ_poly_carrier] polynomial_def by auto
+next
+  interpret UP: domain "K[X]"
+    using univ_poly_is_domain[OF assms(1)] .
+
+  { fix q r
+    assume q: "q \<in> carrier (K[X])" and r: "r \<in> carrier (K[X])" and "[ a, b ] = q \<otimes>\<^bsub>K[X]\<^esub> r"
+    hence not_zero: "q \<noteq> []" "r \<noteq> []"
+      by (metis UP.integral_iff list.distinct(1) univ_poly_zero)+
+    have "degree (q \<otimes>\<^bsub>K[X]\<^esub> r) = degree q + degree r"
+      using not_zero poly_mult_degree_eq[OF assms(1)] q r
+      by (simp add: univ_poly_carrier univ_poly_mult)
+    with sym[OF \<open>[ a, b ] = q \<otimes>\<^bsub>K[X]\<^esub> r\<close>] have "degree q + degree r = 1" and "q \<noteq> []" "r \<noteq> []"
+      using not_zero by auto
+  } note aux_lemma1 = this
+
+  { fix q r
+    assume q: "q \<in> carrier (K[X])" "q \<noteq> []" and r: "r \<in> carrier (K[X])" "r \<noteq> []"
+      and "[ a, b ] = q \<otimes>\<^bsub>K[X]\<^esub> r" and "degree q = 1" and "degree r = 0"
+    hence "length q = Suc (Suc 0)" and "length r = Suc 0"
+    from \<open>length q = Suc (Suc 0)\<close> obtain c d where q_def: "q = [ c, d ]"
+      by (metis length_0_conv length_Cons list.exhaust nat.inject)
+    from \<open>length r = Suc 0\<close> obtain e where r_def: "r = [ e ]"
+      by (metis length_0_conv length_Suc_conv)
+    from \<open>r = [ e ]\<close> and \<open>q = [ c, d ]\<close>
+    have c: "c \<in> K" "c \<noteq> \<zero>" and d: "d \<in> K" and e: "e \<in> K" "e \<noteq> \<zero>"
+      using r q subringE(1)[OF assms(1)] unfolding sym[OF univ_poly_carrier] polynomial_def by auto
+    with sym[OF \<open>[ a, b ] = q \<otimes>\<^bsub>K[X]\<^esub> r\<close>] have "a = c \<otimes> e"
+      using poly_mult_lead_coeff[OF assms(1), of q r]
+      unfolding polynomial_def sym[OF univ_poly_mult[of R K]] r_def q_def by auto
+    obtain inv_a where a: "a \<in> K" and inv_a: "inv_a \<in> K" "a \<otimes> inv_a = \<one>" "inv_a \<otimes> a = \<one>"
+      using assms(2) unfolding Units_def by auto
+    hence "a \<noteq> \<zero>" and "inv_a \<noteq> \<zero>"
+      using subringE(1)[OF assms(1)] integral_iff by auto
+    with \<open>c \<in> K\<close> and \<open>c \<noteq> \<zero>\<close> have in_carrier: "[ c \<otimes> inv_a ] \<in> carrier (K[X])"
+      using subringE(1,6)[OF assms(1)] inv_a integral
+      unfolding sym[OF univ_poly_carrier] polynomial_def
+      by (auto, meson subsetD)
+    moreover have "[ c \<otimes> inv_a ] \<otimes>\<^bsub>K[X]\<^esub> r = [ \<one> ]"
+      using \<open>a = c \<otimes> e\<close> a inv_a c e subsetD[OF subringE(1)[OF assms(1)]]
+      unfolding r_def univ_poly_mult by (auto) (simp add: m_assoc m_lcomm integral_iff)+
+    ultimately have "r \<in> Units (K[X])"
+      using r(1) UP.m_comm[OF in_carrier r(1)] unfolding sym[OF univ_poly_one[of R K]] Units_def by auto
+  } note aux_lemma2 = this
+
+  fix q r
+  assume q: "q \<in> carrier (K[X])" and r: "r \<in> carrier (K[X])" and qr: "[ a, b ] = q \<otimes>\<^bsub>K[X]\<^esub> r"
+  thus "q \<in> Units (K[X]) \<or> r \<in> Units (K[X])"
+    using aux_lemma1[OF q r qr] aux_lemma2[of q r] aux_lemma2[of r q] UP.m_comm add_is_1 by auto
+qed
+
+(* MOVE to Polynomial_Divisibility.thy *)
+lemma (in domain) degree_one_imp_pirreducible:
+  assumes "subfield K R" and "p \<in> carrier (K[X])" and "degree p = 1"
+  shows "pirreducible K p"
+proof -
+  from \<open>degree p = 1\<close> have "length p = Suc (Suc 0)"
+    by simp
+  then obtain a b where p: "p = [ a, b ]"
+    by (metis length_0_conv length_Suc_conv)
+  with \<open>p \<in> carrier (K[X])\<close> show ?thesis
+    using subring_degree_one_imp_pirreducible[OF subfieldE(1)[OF assms(1)], of a b]
+          subfield.subfield_Units[OF assms(1)]
+    unfolding sym[OF univ_poly_carrier] polynomial_def by auto
+qed
+
+(* MOVE to Polynomial_Divisibility.thy *)
+lemma (in ring) degree_oneE[elim]:
+  assumes "p \<in> carrier (K[X])" and "degree p = 1"
+    and "\<And>a b. \<lbrakk> a \<in> K; a \<noteq> \<zero>; b \<in> K; p = [ a, b ] \<rbrakk> \<Longrightarrow> P"
+  shows P
+proof -
+  from \<open>degree p = 1\<close> have "length p = Suc (Suc 0)"
+    by simp
+  then obtain a b where "p = [ a, b ]"
+    by (metis length_0_conv length_Cons nat.inject neq_Nil_conv)
+  with \<open>p \<in> carrier (K[X])\<close> have "a \<in> K" and "a \<noteq> \<zero>" and "b \<in> K"
+    unfolding sym[OF univ_poly_carrier] polynomial_def by auto
+  with \<open>p = [ a, b ]\<close> show ?thesis
+    using assms(3) by simp
+qed
+
+(* MOVE to Polynomial_Divisibility.thy *)
+lemma (in domain) subring_degree_one_associatedI:
+  assumes "subring K R" and "a \<in> K" "a' \<in> K" and "b \<in> K" and "a \<otimes> a' = \<one>"
+  shows "[ a , b ] \<sim>\<^bsub>K[X]\<^esub> [ \<one>, a' \<otimes> b ]"
+proof -
+  from \<open>a \<otimes> a' = \<one>\<close> have not_zero: "a \<noteq> \<zero>" "a' \<noteq> \<zero>"
+    using subringE(1)[OF assms(1)] assms(2-3) by auto
+  hence "[ a, b ] = [ a ] \<otimes>\<^bsub>K[X]\<^esub> [ \<one>, a' \<otimes> b ]"
+    using assms(2-4)[THEN subsetD[OF subringE(1)[OF assms(1)]]] assms(5) m_assoc
+    unfolding univ_poly_mult by fastforce
+  moreover have "[ a, b ] \<in> carrier (K[X])" and "[ \<one>, a' \<otimes> b ] \<in> carrier (K[X])"
+    using subringE(1,3,6)[OF assms(1)] not_zero one_not_zero assms
+    unfolding sym[OF univ_poly_carrier] polynomial_def by auto
+  moreover have "[ a ] \<in> Units (K[X])"
+  proof -
+    from \<open>a \<noteq> \<zero>\<close> and \<open>a' \<noteq> \<zero>\<close> have "[ a ] \<in> carrier (K[X])" and "[ a' ] \<in> carrier (K[X])"
+      using assms(2-3) unfolding sym[OF univ_poly_carrier] polynomial_def by auto
+    moreover have "a' \<otimes> a = \<one>"
+      using subsetD[OF subringE(1)[OF assms(1)]] assms m_comm by simp
+    hence "[ a ] \<otimes>\<^bsub>K[X]\<^esub> [ a' ] = [ \<one> ]" and "[ a' ] \<otimes>\<^bsub>K[X]\<^esub> [ a ] = [ \<one> ]"
+      using assms unfolding univ_poly_mult by auto
+    ultimately show ?thesis
+      unfolding sym[OF univ_poly_one[of R K]] Units_def by blast
+  qed
+  ultimately show ?thesis
+    using domain.ring_associated_iff[OF univ_poly_is_domain[OF assms(1)]] by blast
+qed
+
+(* MOVE to Polynomial_Divisibility.thy *)
+lemma (in domain) degree_one_associatedI:
+  assumes "subfield K R" and "p \<in> carrier (K[X])" and "degree p = 1"
+  shows "p \<sim>\<^bsub>K[X]\<^esub> [ \<one>, inv (lead_coeff p) \<otimes> (const_term p) ]"
+proof -
+  from \<open>p \<in> carrier (K[X])\<close> and \<open>degree p = 1\<close>
+  obtain a b where "p = [ a, b ]" and "a \<in> K" "a \<noteq> \<zero>" and "b \<in> K"
+    by auto
+  thus ?thesis
+    using subring_degree_one_associatedI[OF subfieldE(1)[OF assms(1)]]
+          subfield_m_inv[OF assms(1)] subsetD[OF subfieldE(3)[OF assms(1)]]
+    unfolding const_term_def
+    by auto
+qed
+
+lemma (in domain) monic_degree_one_roots:
+  assumes "a \<in> carrier R" shows "roots [ \<one> , \<ominus> a ] = {# a #}"
+proof -
+  let ?p = "[ \<one> , \<ominus> a ]"
+
+  interpret UP: domain "poly_ring R"
+    using univ_poly_is_domain[OF carrier_is_subring] .
+
+  from \<open>a \<in> carrier R\<close> have in_carrier: "?p \<in> carrier (poly_ring R)"
+    unfolding sym[OF univ_poly_carrier] polynomial_def by simp
+  show ?thesis
+  proof (rule subset_mset.antisym)
+    show "{# a #} \<subseteq># roots ?p"
+      using roots_mem_iff_is_root[OF in_carrier]
+            monic_degree_one_root_condition[OF assms]
+      by simp
+  next
+    show "roots ?p \<subseteq># {# a #}"
+    proof (rule mset_subset_eqI, auto)
+      fix b assume "a \<noteq> b" thus "count (roots ?p) b = 0"
+        using alg_mult_gt_zero_iff_is_root[OF in_carrier]
+              monic_degree_one_root_condition[OF assms]
+        unfolding sym[OF alg_mult_eq_count_roots[OF in_carrier]]
+        by fastforce
+    next
+      have "(?p [^]\<^bsub>poly_ring R\<^esub> (alg_mult ?p a)) pdivides ?p"
+        using le_alg_mult_imp_pdivides[OF assms in_carrier] by simp
+      hence "degree (?p [^]\<^bsub>poly_ring R\<^esub> (alg_mult ?p a)) \<le> degree ?p"
+        using pdivides_imp_degree_le[OF carrier_is_subring, of _ ?p] in_carrier by auto
+      thus "count (roots ?p) a \<le> Suc 0"
+        using polynomial_pow_degree[OF in_carrier]
+        unfolding sym[OF alg_mult_eq_count_roots[OF in_carrier]]
+        by auto
+    qed
+  qed
+qed
+
+lemma (in domain) degree_one_roots:
+  assumes "a \<in> carrier R" "a' \<in> carrier R" and "b \<in> carrier R" and "a \<otimes> a' = \<one>"
+  shows "roots [ a , b ] = {# \<ominus> (a' \<otimes> b) #}"
+proof -
+  have "[ a, b ] \<in> carrier (poly_ring R)" and "[ \<one>, a' \<otimes> b ] \<in> carrier (poly_ring R)"
+    using assms unfolding sym[OF univ_poly_carrier] polynomial_def by auto
+  thus ?thesis
+    using subring_degree_one_associatedI[OF carrier_is_subring assms] assms
+          monic_degree_one_roots associated_polynomials_imp_same_roots
+    by (metis add.inv_closed local.minus_minus m_closed)
+qed
+
+lemma (in field) degree_one_imp_singleton_roots:
+  assumes "p \<in> carrier (poly_ring R)" and "degree p = 1"
+  shows "roots p = {# \<ominus> (inv (lead_coeff p) \<otimes> (const_term p)) #}"
+proof -
+  from \<open>p \<in> carrier (poly_ring R)\<close> and \<open>degree p = 1\<close>
+  obtain a b where "p = [ a, b ]" and "a \<in> carrier R" "a \<noteq> \<zero>" and "b \<in> carrier R"
+    by auto
+  thus ?thesis
+    using degree_one_roots[of a "inv a" b]
+    by (auto simp add: const_term_def field_Units)
+qed
+
+lemma (in field) degree_one_imp_splitted:
+  assumes "p \<in> carrier (poly_ring R)" and "degree p = 1" shows "splitted p"
+  using degree_one_imp_singleton_roots[OF assms] assms(2) unfolding splitted_def by simp
+
+lemma (in field) no_roots_imp_same_roots:
+  assumes "p \<in> carrier (poly_ring R)" "p \<noteq> []" and "q \<in> carrier (poly_ring R)"
+  shows "roots p = {#} \<Longrightarrow> roots (p \<otimes>\<^bsub>poly_ring R\<^esub> q) = roots q"
+proof -
+  interpret UP: domain "poly_ring R"
+    using univ_poly_is_domain[OF carrier_is_subring] .
+
+  assume no_roots: "roots p = {#}" show "roots (p \<otimes>\<^bsub>poly_ring R\<^esub> q) = roots q"
+  proof (intro subset_mset.antisym)
+    have pdiv: "q pdivides (p \<otimes>\<^bsub>poly_ring R\<^esub> q)"
+      using UP.divides_prod_l assms unfolding pdivides_def by blast
+    show "roots q \<subseteq># roots (p \<otimes>\<^bsub>poly_ring R\<^esub> q)"
+      using pdivides_imp_roots_incl[OF _ _ _ pdiv] assms
+            degree_zero_imp_empty_roots[OF assms(3)]
+      by (cases "q = []", auto, metis UP.l_null UP.m_rcancel UP.zero_closed univ_poly_zero)
+  next
+    show "roots (p \<otimes>\<^bsub>poly_ring R\<^esub> q) \<subseteq># roots q"
+    proof (cases "p \<otimes>\<^bsub>poly_ring R\<^esub> q = []")
+      case True thus ?thesis
+        using degree_zero_imp_empty_roots[OF UP.m_closed[OF assms(1,3)]] by simp
+    next
+      case False with \<open>p \<noteq> []\<close> have q_not_zero: "q \<noteq> []"
+        by (metis UP.r_null assms(1) univ_poly_zero)
+      show ?thesis
+      proof (rule roots_inclI[OF UP.m_closed[OF assms(1,3)] assms(3) q_not_zero])
+        fix a assume a: "a \<in> carrier R"
+        hence "\<not> ([ \<one>, \<ominus> a ] pdivides p)"
+          using assms(1-2) no_roots pdivides_imp_is_root roots_mem_iff_is_root[of p] by auto
+        moreover have in_carrier: "[ \<one>, \<ominus> a ] \<in> carrier (poly_ring R)"
+          using a unfolding sym[OF univ_poly_carrier] polynomial_def by auto
+        hence "pirreducible (carrier R) [ \<one>, \<ominus> a ]"
+          using degree_one_imp_pirreducible[OF carrier_is_subfield] by simp
+        moreover
+        have "([ \<one>, \<ominus> a ] [^]\<^bsub>poly_ring R\<^esub> (alg_mult (p \<otimes>\<^bsub>poly_ring R\<^esub> q) a)) pdivides (p \<otimes>\<^bsub>poly_ring R\<^esub> q)"
+          using le_alg_mult_imp_pdivides[OF a UP.m_closed, of p q] assms by simp
+        ultimately show "([ \<one>, \<ominus> a ] [^]\<^bsub>poly_ring R\<^esub> (alg_mult (p \<otimes>\<^bsub>poly_ring R\<^esub> q) a)) pdivides q"
+          using pirreducible_pow_pdivides_iff[OF carrier_is_subfield in_carrier] assms by auto
+      qed
+    qed
+  qed
+qed
+
+lemma (in field) poly_mult_degree_one_monic_imp_same_roots:
+  assumes "a \<in> carrier R" and "p \<in> carrier (poly_ring R)" "p \<noteq> []"
+  shows "roots ([ \<one>, \<ominus> a ] \<otimes>\<^bsub>poly_ring R\<^esub> p) = add_mset a (roots p)"
+proof -
+  interpret UP: domain "poly_ring R"
+    using univ_poly_is_domain[OF carrier_is_subring] .
+
+  from \<open>a \<in> carrier R\<close> have in_carrier: "[ \<one>, \<ominus> a ] \<in> carrier (poly_ring R)"
+    unfolding sym[OF univ_poly_carrier] polynomial_def by simp
+
+  show ?thesis
+  proof (intro subset_mset.antisym[OF roots_inclI' mset_subset_eqI])
+    show "([ \<one>, \<ominus> a ] \<otimes>\<^bsub>poly_ring R\<^esub> p) \<in> carrier (poly_ring R)"
+      using in_carrier assms(2) by simp
+  next
+    fix b assume b: "b \<in> carrier R" and "[ \<one>, \<ominus> a ] \<otimes>\<^bsub>poly_ring R\<^esub> p \<noteq> []"
+    hence not_zero: "p \<noteq> []"
+      unfolding univ_poly_def by auto
+    from \<open>b \<in> carrier R\<close> have in_carrier':  "[ \<one>, \<ominus> b ] \<in> carrier (poly_ring R)"
+      unfolding sym[OF univ_poly_carrier] polynomial_def by simp
+    show "alg_mult ([ \<one>, \<ominus> a ] \<otimes>\<^bsub>poly_ring R\<^esub> p) b \<le> count (add_mset a (roots p)) b"
+    proof (cases "a = b")
+      case False
+      hence "\<not> [ \<one>, \<ominus> b ] pdivides [ \<one>, \<ominus> a ]"
+        using assms(1) b monic_degree_one_root_condition pdivides_imp_is_root by blast
+      moreover have "pirreducible (carrier R) [ \<one>, \<ominus> b ]"
+        using degree_one_imp_pirreducible[OF carrier_is_subfield in_carrier'] by simp
+      ultimately
+      have "[ \<one>, \<ominus> b ] [^]\<^bsub>poly_ring R\<^esub> (alg_mult ([ \<one>, \<ominus> a ] \<otimes>\<^bsub>poly_ring R\<^esub> p) b) pdivides p"
+        using le_alg_mult_imp_pdivides[OF b UP.m_closed, of _ p] assms(2) in_carrier
+              pirreducible_pow_pdivides_iff[OF carrier_is_subfield in_carrier' in_carrier, of p]
+        by auto
+      with \<open>a \<noteq> b\<close> show ?thesis
+        using alg_mult_eq_count_roots[OF assms(2)] alg_multE(2)[OF b assms(2) not_zero] by auto
+    next
+      case True
+      have "[ \<one>, \<ominus> a ] pdivides ([ \<one>, \<ominus> a ] \<otimes>\<^bsub>poly_ring R\<^esub> p)"
+        using dividesI[OF assms(2)] unfolding pdivides_def by auto
+      with \<open>[ \<one>, \<ominus> a ] \<otimes>\<^bsub>poly_ring R\<^esub> p \<noteq> []\<close>
+      have "alg_mult ([ \<one>, \<ominus> a ] \<otimes>\<^bsub>poly_ring R\<^esub> p) a \<ge> Suc 0"
+        using alg_multE(2)[of a _ "Suc 0"] in_carrier assms by auto
+      then obtain m where m: "alg_mult ([ \<one>, \<ominus> a ] \<otimes>\<^bsub>poly_ring R\<^esub> p) a = Suc m"
+        using Suc_le_D by blast
+      hence "([ \<one>, \<ominus> a ] \<otimes>\<^bsub>poly_ring R\<^esub> ([ \<one>, \<ominus> a ] [^]\<^bsub>poly_ring R\<^esub> m)) pdivides
+             ([ \<one>, \<ominus> a ] \<otimes>\<^bsub>poly_ring R\<^esub> p)"
+        using le_alg_mult_imp_pdivides[OF _ UP.m_closed, of a _ p]
+              in_carrier assms UP.nat_pow_Suc2 by force
+      hence "([ \<one>, \<ominus> a ] [^]\<^bsub>poly_ring R\<^esub> m) pdivides p"
+        using UP.mult_divides in_carrier assms(2)
+        unfolding univ_poly_zero pdivides_def factor_def
+        by (simp add: UP.m_assoc UP.m_lcancel univ_poly_zero)
+      with \<open>a = b\<close> show ?thesis
+        using alg_mult_eq_count_roots assms in_carrier UP.nat_pow_Suc2
+              alg_multE(2)[OF assms(1) _ not_zero] m
+        by auto
+    qed
+  next
+    fix b
+    have not_zero: "[ \<one>, \<ominus> a ] \<otimes>\<^bsub>poly_ring R\<^esub> p \<noteq> []"
+      using assms in_carrier univ_poly_zero[of R] UP.integral by auto
+
+    show "count (add_mset a (roots p)) b \<le> count (roots ([\<one>, \<ominus> a] \<otimes>\<^bsub>poly_ring R\<^esub> p)) b"
+    proof (cases "a = b")
+      case True
+      have "([ \<one>, \<ominus> a ] \<otimes>\<^bsub>poly_ring R\<^esub> ([ \<one>, \<ominus> a ] [^]\<^bsub>poly_ring R\<^esub> (alg_mult p a))) pdivides
+            ([ \<one>, \<ominus> a ] \<otimes>\<^bsub>poly_ring R\<^esub> p)"
+        using UP.divides_mult[OF _ in_carrier] le_alg_mult_imp_pdivides[OF assms(1,2)] in_carrier assms
+        by (auto simp add: pdivides_def)
+      with \<open>a = b\<close> show ?thesis
+        using alg_mult_eq_count_roots assms in_carrier UP.nat_pow_Suc2
+              alg_multE(2)[OF assms(1) _ not_zero]
+        by auto
+    next
+      case False
+      have "p pdivides ([ \<one>, \<ominus> a ] \<otimes>\<^bsub>poly_ring R\<^esub> p)"
+        using dividesI[OF in_carrier] UP.m_comm in_carrier assms unfolding pdivides_def by auto
+      thus ?thesis
+        using False pdivides_imp_roots_incl assms in_carrier not_zero
+    qed
+  qed
+qed
+
+lemma (in domain) not_empty_rootsE[elim]:
+  assumes "p \<in> carrier (poly_ring R)" and "roots p \<noteq> {#}"
+    and "\<And>a. \<lbrakk> a \<in> carrier R; a \<in># roots p;
+               [ \<one>, \<ominus> a ] \<in> carrier (poly_ring R); [ \<one>, \<ominus> a ] pdivides p \<rbrakk> \<Longrightarrow> P"
+  shows P
+proof -
+  from \<open>roots p \<noteq> {#}\<close> obtain a where "a \<in># roots p"
+    by blast
+  with \<open>p \<in> carrier (poly_ring R)\<close> have "[ \<one>, \<ominus> a ] pdivides p"
+    and "[ \<one>, \<ominus> a ] \<in> carrier (poly_ring R)" and "a \<in> carrier R"
+    using is_root_imp_pdivides[of p] roots_mem_iff_is_root[of p] is_root_def[of p a]
+    unfolding sym[OF univ_poly_carrier] polynomial_def by auto
+  with \<open>a \<in># roots p\<close> show ?thesis
+    using assms(3)[of a] by auto
+qed
+
+(* REPLACE th following lemmas on Divisibility.thy ============= *)
+(* the only difference is the locale                             *)
+lemma (in monoid) mult_cong_r:
+  assumes "b \<sim> b'" "a \<in> carrier G"  "b \<in> carrier G"  "b' \<in> carrier G"
+  shows "a \<otimes> b \<sim> a \<otimes> b'"
+  by (meson assms associated_def divides_mult_lI)
+
+lemma (in comm_monoid) mult_cong_l:
+  assumes "a \<sim> a'" "a \<in> carrier G"  "a' \<in> carrier G"  "b \<in> carrier G"
+  shows "a \<otimes> b \<sim> a' \<otimes> b"
+  using assms m_comm mult_cong_r by auto
+(* ============================================================= *)
+
+lemma (in field) associated_polynomials_imp_same_roots:
+  assumes "p \<in> carrier (poly_ring R)" "p \<noteq> []" and "q \<in> carrier (poly_ring R)" "q \<noteq> []"
+  shows "roots (p \<otimes>\<^bsub>poly_ring R\<^esub> q) = roots p + roots q"
+proof -
+  interpret UP: domain "poly_ring R"
+    using univ_poly_is_domain[OF carrier_is_subring] .
+  from assms show ?thesis
+  proof (induction "degree p" arbitrary: p rule: less_induct)
+    case less show ?case
+    proof (cases "roots p = {#}")
+      case True thus ?thesis
+        using no_roots_imp_same_roots[of p q] less by simp
+    next
+      case False with \<open>p \<in> carrier (poly_ring R)\<close>
+      obtain a where a: "a \<in> carrier R" and "a \<in># roots p" and pdiv: "[ \<one>, \<ominus> a ] pdivides p"
+          and in_carrier: "[ \<one>, \<ominus> a ] \<in> carrier (poly_ring R)"
+        by blast
+      show ?thesis
+      proof (cases "degree p = 1")
+        case True with \<open>p \<in> carrier (poly_ring R)\<close>
+        obtain b c where p: "p = [ b, c ]" and b: "b \<in> carrier R" "b \<noteq> \<zero>" and c: "c \<in> carrier R"
+          by auto
+        with \<open>a \<in># roots p\<close> have roots: "roots p = {# a #}" and a: "\<ominus> a = inv b \<otimes> c" "a \<in> carrier R"
+          and lead: "lead_coeff p = b" and const: "const_term p = c"
+          using degree_one_imp_singleton_roots[of p] less(2) field_Units
+          unfolding const_term_def by auto
+        hence "(p \<otimes>\<^bsub>poly_ring R\<^esub> q) \<sim>\<^bsub>poly_ring R\<^esub> ([ \<one>, \<ominus> a ] \<otimes>\<^bsub>poly_ring R\<^esub> q)"
+          using UP.mult_cong_l[OF degree_one_associatedI[OF carrier_is_subfield _ True]] less(2,4)
+        hence "roots (p \<otimes>\<^bsub>poly_ring R\<^esub> q) = roots ([ \<one>, \<ominus> a ] \<otimes>\<^bsub>poly_ring R\<^esub> q)"
+          using associated_polynomials_imp_same_roots in_carrier less(2,4) unfolding a by simp
+        thus ?thesis
+          unfolding poly_mult_degree_one_monic_imp_same_roots[OF a(2) less(4,5)] roots by simp
+      next
+        case False
+        from \<open>[ \<one>, \<ominus> a ] pdivides p\<close>
+        obtain r where p: "p = [ \<one>, \<ominus> a ] \<otimes>\<^bsub>poly_ring R\<^esub> r" and r: "r \<in> carrier (poly_ring R)"
+          unfolding pdivides_def by auto
+        with \<open>p \<noteq> []\<close> have not_zero: "r \<noteq> []"
+          using in_carrier univ_poly_zero[of R "carrier R"] UP.integral_iff by auto
+        with \<open>p = [ \<one>, \<ominus> a ] \<otimes>\<^bsub>poly_ring R\<^esub> r\<close> have deg: "degree p = Suc (degree r)"
+          using poly_mult_degree_eq[OF carrier_is_subring, of _ r] in_carrier r
+          unfolding univ_poly_carrier sym[OF univ_poly_mult[of R "carrier R"]] by auto
+        with \<open>r \<noteq> []\<close> and \<open>q \<noteq> []\<close> have "r \<otimes>\<^bsub>poly_ring R\<^esub> q \<noteq> []"
+          using in_carrier univ_poly_zero[of R "carrier R"] UP.integral less(4) r by auto
+        hence "roots (p \<otimes>\<^bsub>poly_ring R\<^esub> q) = add_mset a (roots (r \<otimes>\<^bsub>poly_ring R\<^esub> q))"
+          using poly_mult_degree_one_monic_imp_same_roots[OF a UP.m_closed[OF r less(4)]]
+                UP.m_assoc[OF in_carrier r less(4)] p by auto
+        also have " ... = add_mset a (roots r + roots q)"
+          using less(1)[OF _ r not_zero less(4-5)] deg by simp
+        also have " ... = (add_mset a (roots r)) + roots q"
+          by simp
+        also have " ... = roots p + roots q"
+          using poly_mult_degree_one_monic_imp_same_roots[OF a r not_zero] p by simp
+        finally show ?thesis .
+      qed
+    qed
+  qed
+qed
+
+lemma (in field) size_roots_le_degree:
+  assumes "p \<in> carrier (poly_ring R)" shows "size (roots p) \<le> degree p"
+  using assms
+proof (induction "degree p" arbitrary: p rule: less_induct)
+  case less show ?case
+  proof (cases "roots p = {#}", simp)
+    interpret UP: domain "poly_ring R"
+      using univ_poly_is_domain[OF carrier_is_subring] .
+
+    case False with \<open>p \<in> carrier (poly_ring R)\<close>
+    obtain a where a: "a \<in> carrier R" and "a \<in># roots p" and "[ \<one>, \<ominus> a ] pdivides p"
+      and in_carrier: "[ \<one>, \<ominus> a ] \<in> carrier (poly_ring R)"
+      by blast
+    then obtain q where p: "p = [ \<one>, \<ominus> a ] \<otimes>\<^bsub>poly_ring R\<^esub> q" and q: "q \<in> carrier (poly_ring R)"
+      unfolding pdivides_def by auto
+    with \<open>a \<in># roots p\<close> have "p \<noteq> []"
+      using degree_zero_imp_empty_roots[OF less(2)] by auto
+    hence not_zero: "q \<noteq> []"
+      using in_carrier univ_poly_zero[of R "carrier R"] UP.integral_iff p by auto
+    hence "degree p = Suc (degree q)"
+      using poly_mult_degree_eq[OF carrier_is_subring, of _ q] in_carrier p q
+      unfolding univ_poly_carrier sym[OF univ_poly_mult[of R "carrier R"]] by auto
+    with \<open>q \<noteq> []\<close> show ?thesis
+      using poly_mult_degree_one_monic_imp_same_roots[OF a q] p less(1)[OF _ q]
+      by (metis Suc_le_mono lessI size_add_mset)
+  qed
+qed
+
+(* MOVE to Divisibility.thy ======== *)
+lemma divides_irreducible_condition:
+  assumes "irreducible G r" and "a \<in> carrier G"
+  shows "a divides\<^bsub>G\<^esub> r \<Longrightarrow> a \<in> Units G \<or> a \<sim>\<^bsub>G\<^esub> r"
+  using assms unfolding irreducible_def properfactor_def associated_def
+  by (cases "r divides\<^bsub>G\<^esub> a", auto)
+
+(* MOVE to Polynomial_Divisibility.thy ======== *)
+lemma (in ring) divides_pirreducible_condition:
+  assumes "pirreducible K q" and "p \<in> carrier (K[X])"
+  shows "p divides\<^bsub>K[X]\<^esub> q \<Longrightarrow> p \<in> Units (K[X]) \<or> p \<sim>\<^bsub>K[X]\<^esub> q"
+  using divides_irreducible_condition[of "K[X]" q p] assms
+  unfolding ring_irreducible_def by auto
+
+lemma (in domain) pirreducible_roots:
+  assumes "p \<in> carrier (poly_ring R)" and "pirreducible (carrier R) p" and "degree p \<noteq> 1"
+  shows "roots p = {#}"
+proof (rule ccontr)
+  assume "roots p \<noteq> {#}" with \<open>p \<in> carrier (poly_ring R)\<close>
+  obtain a where a: "a \<in> carrier R" and "a \<in># roots p" and "[ \<one>, \<ominus> a ] pdivides p"
+    and in_carrier: "[ \<one>, \<ominus> a ] \<in> carrier (poly_ring R)"
+    by blast
+  hence "[ \<one>, \<ominus> a ] \<sim>\<^bsub>poly_ring R\<^esub> p"
+    using divides_pirreducible_condition[OF assms(2) in_carrier]
+          univ_poly_units_incl[OF carrier_is_subring]
+    unfolding pdivides_def by auto
+  hence "degree p = 1"
+    using associated_polynomials_imp_same_length[OF carrier_is_subring in_carrier assms(1)] by auto
+  with \<open>degree p \<noteq> 1\<close> show False ..
+qed
+
+lemma (in field) pirreducible_imp_not_splitted:
+  assumes "p \<in> carrier (poly_ring R)" and "pirreducible (carrier R) p" and "degree p \<noteq> 1"
+  shows "\<not> splitted p"
+  using pirreducible_roots[of p] pirreducible_degree[OF carrier_is_subfield, of p] assms
+
+lemma (in field)
+  assumes "p \<in> carrier (poly_ring R)" and "q \<in> carrier (poly_ring R)"
+    and "pirreducible (carrier R) p" and "degree p \<noteq> 1"
+  shows "roots (p \<otimes>\<^bsub>poly_ring R\<^esub> q) = roots q"
+  using no_roots_imp_same_roots[of p q] pirreducible_roots[of p] assms
+  unfolding ring_irreducible_def univ_poly_zero by auto
+
+lemma (in field) trivial_factors_imp_splitted:
+  assumes "p \<in> carrier (poly_ring R)"
+    and "\<And>q. \<lbrakk> q \<in> carrier (poly_ring R); pirreducible (carrier R) q; q pdivides p \<rbrakk> \<Longrightarrow> degree q \<le> 1"
+  shows "splitted p"
+  using assms
+proof (induction "degree p" arbitrary: p rule: less_induct)
+  interpret UP: principal_domain "poly_ring R"
+    using univ_poly_is_principal[OF carrier_is_subfield] .
+  case less show ?case
+  proof (cases "degree p = 0", simp add: degree_zero_imp_splitted[OF less(2)])
+    case False show ?thesis
+    proof (cases "roots p = {#}")
+      case True
+      from \<open>degree p \<noteq> 0\<close> have "p \<notin> Units (poly_ring R)" and "p \<in> carrier (poly_ring R) - { [] }"
+        using univ_poly_units'[OF carrier_is_subfield, of p] less(2) by auto
+      then obtain q where "q \<in> carrier (poly_ring R)" "pirreducible (carrier R) q" and "q pdivides p"
+        using UP.exists_irreducible_divisor[of p] unfolding univ_poly_zero pdivides_def by auto
+      with \<open>degree p \<noteq> 0\<close> have "roots p \<noteq> {#}"
+        using degree_one_imp_singleton_roots[OF _ , of q] less(3)[of q]
+              pdivides_imp_roots_incl[OF _ less(2), of q]
+              pirreducible_degree[OF carrier_is_subfield, of q]
+        by force
+      from \<open>roots p = {#}\<close> and \<open>roots p \<noteq> {#}\<close> have False
+        by simp
+      thus ?thesis ..
+    next
+      case False with \<open>p \<in> carrier (poly_ring R)\<close>
+      obtain a where a: "a \<in> carrier R" and "a \<in># roots p" and "[ \<one>, \<ominus> a ] pdivides p"
+        and in_carrier: "[ \<one>, \<ominus> a ] \<in> carrier (poly_ring R)"
+        by blast
+      then obtain q where p: "p = [ \<one>, \<ominus> a ] \<otimes>\<^bsub>poly_ring R\<^esub> q" and q: "q \<in> carrier (poly_ring R)"
+        unfolding pdivides_def by blast
+      with \<open>degree p \<noteq> 0\<close> have "p \<noteq> []"
+        by auto
+      with \<open>p = [ \<one>, \<ominus> a ] \<otimes>\<^bsub>poly_ring R\<^esub> q\<close> have "q \<noteq> []"
+        using in_carrier q unfolding sym[OF univ_poly_zero[of R "carrier R"]] by auto
+      with \<open>p = [ \<one>, \<ominus> a ] \<otimes>\<^bsub>poly_ring R\<^esub> q\<close> and \<open>p \<noteq> []\<close> have "degree p = Suc (degree q)"
+        using poly_mult_degree_eq[OF carrier_is_subring] in_carrier q
+        unfolding univ_poly_carrier sym[OF univ_poly_mult[of R "carrier R"]] by auto
+      moreover have "q pdivides p"
+        using p dividesI[OF in_carrier] UP.m_comm[OF in_carrier q] by (auto simp add: pdivides_def)
+      hence "degree r = 1" if "r \<in> carrier (poly_ring R)" and "pirreducible (carrier R) r"
+        and "r pdivides q" for r
+        using less(3)[OF that(1-2)] UP.divides_trans[OF _ _ that(1), of q p] that(3)
+              pirreducible_degree[OF carrier_is_subfield that(1-2)]
+        by (auto simp add: pdivides_def)
+      ultimately have "splitted q"
+        using less(1)[OF _ q] by auto
+      with \<open>degree p = Suc (degree q)\<close> and \<open>q \<noteq> []\<close> show ?thesis
+        using poly_mult_degree_one_monic_imp_same_roots[OF a q]
+        unfolding sym[OF p] splitted_def
+        by simp
+    qed
+  qed
+qed
+
+lemma (in field) pdivides_imp_splitted:
+  assumes "p \<in> carrier (poly_ring R)" and "q \<in> carrier (poly_ring R)" "q \<noteq> []" and "splitted q"
+  shows "p pdivides q \<Longrightarrow> splitted p"
+proof (cases "p = []")
+  case True thus ?thesis
+    using degree_zero_imp_splitted[OF assms(1)] by simp
+next
+  interpret UP: principal_domain "poly_ring R"
+    using univ_poly_is_principal[OF carrier_is_subfield] .
+
+  case False
+  assume "p pdivides q"
+  then obtain b where b: "b \<in> carrier (poly_ring R)" and q: "q = p \<otimes>\<^bsub>poly_ring R\<^esub> b"
+    unfolding pdivides_def by auto
+  with \<open>q \<noteq> []\<close> have "p \<noteq> []" and "b \<noteq> []"
+    using assms UP.integral_iff[of p b] unfolding sym[OF univ_poly_zero[of R "carrier R"]] by auto
+  hence "degree p + degree b = size (roots p) + size (roots b)"
+    using associated_polynomials_imp_same_roots[of p b] assms b q splitted_def
+          poly_mult_degree_eq[OF carrier_is_subring,of p b]
+    unfolding univ_poly_carrier sym[OF univ_poly_mult[of R "carrier R"]]
+    by auto
+  moreover have "size (roots p) \<le> degree p" and "size (roots b) \<le> degree b"
+    using size_roots_le_degree assms(1) b by auto
+  ultimately show ?thesis
+    unfolding splitted_def by linarith
+qed
+
+lemma (in field) splitted_imp_trivial_factors:
+  assumes "p \<in> carrier (poly_ring R)" "p \<noteq> []" and "splitted p"
+  shows "\<And>q. \<lbrakk> q \<in> carrier (poly_ring R); pirreducible (carrier R) q; q pdivides p \<rbrakk> \<Longrightarrow> degree q = 1"
+  using pdivides_imp_splitted[OF _ assms] pirreducible_imp_not_splitted
+  by auto
+
lemma (in field) exists_root:
assumes "M \<in> extensions" and "\<And>L. \<lbrakk> L \<in> extensions; M \<lesssim> L \<rbrakk> \<Longrightarrow> law_restrict L = law_restrict M"
-    and "P \<in> carrier (poly_ring R)" and "degree P > 0"
-  shows "\<exists>x \<in> carrier M. (ring.eval M) (\<sigma> P) x = \<zero>\<^bsub>M\<^esub>"
+    and "P \<in> carrier (poly_ring R)"
+  shows "(ring.splitted M) (\<sigma> P)"
proof (rule ccontr)
from \<open>M \<in> extensions\<close> interpret M: field M + Hom: ring_hom_ring R M "indexed_const"
using ring_hom_ringI2[OF ring_axioms field.is_ring] unfolding extensions_def by auto
interpret UP: principal_domain "poly_ring M"
using M.univ_poly_is_principal[OF M.carrier_is_subfield] .

-  assume no_roots: "\<not> (\<exists>x \<in> carrier M. M.eval (\<sigma> P) x = \<zero>\<^bsub>M\<^esub>)"
+  assume not_splitted: "\<not> (ring.splitted M) (\<sigma> P)"
have "(\<sigma> P) \<in> carrier (poly_ring M)"
using polynomial_hom[OF Hom.homh field_axioms M.field_axioms assms(3)] unfolding \<sigma>_def by simp
-  moreover have "(\<sigma> P) \<notin> Units (poly_ring M)" and "(\<sigma> P) \<noteq> \<zero>\<^bsub>poly_ring M\<^esub>"
-    using assms(4) unfolding M.univ_poly_carrier_units \<sigma>_def univ_poly_zero by auto
-  ultimately obtain Q
+  then obtain Q
where Q: "Q \<in> carrier (poly_ring M)" "pirreducible\<^bsub>M\<^esub> (carrier M) Q" "Q pdivides\<^bsub>M\<^esub> (\<sigma> P)"
-    using UP.exists_irreducible_divisor[of "\<sigma> P"] unfolding pdivides_def by blast
+      and degree_gt: "degree Q > 1"
+    using M.trivial_factors_imp_splitted[of "\<sigma> P"] not_splitted by force
+
+  from \<open>(\<sigma> P) \<in> carrier (poly_ring M)\<close> have "(\<sigma> P) \<noteq> []"
+    using M.degree_zero_imp_splitted[of "\<sigma> P"] not_splitted unfolding \<sigma>_def by auto

-  have hyps:
+  have "\<exists>i. \<forall>\<P> \<in> carrier M. index_free \<P> (P, i)"
+  proof (rule ccontr)
+    assume "\<nexists>i. \<forall>\<P> \<in> carrier M. index_free \<P> (P, i)"
+    then have "\<X>\<^bsub>(P, i)\<^esub> \<in> carrier M" and "(ring.eval M) (\<sigma> P) \<X>\<^bsub>(P, i)\<^esub> = \<zero>\<^bsub>M\<^esub>" for i
+      using assms(1,3) unfolding extensions_def by blast+
+    with \<open>(\<sigma> P) \<noteq> []\<close> have "((\<lambda>i :: nat. \<X>\<^bsub>(P, i)\<^esub>) ` UNIV) \<subseteq> { a. (ring.is_root M) (\<sigma> P) a }"
+      unfolding M.is_root_def by auto
+    moreover have "inj (\<lambda>i :: nat. \<X>\<^bsub>(P, i)\<^esub>)"
+      unfolding indexed_var_def indexed_const_def indexed_pmult_def inj_def
+      by (metis (no_types, lifting) add_mset_eq_singleton_iff diff_single_eq_union
+                                    multi_member_last prod.inject zero_not_one)
+    hence "infinite ((\<lambda>i :: nat. \<X>\<^bsub>(P, i)\<^esub>) ` UNIV)"
+      unfolding infinite_iff_countable_subset by auto
+    ultimately have "infinite { a. (ring.is_root M) (\<sigma> P) a }"
+      using finite_subset by auto
+    with \<open>(\<sigma> P) \<in> carrier (poly_ring M)\<close> show False
+      using M.finite_number_of_roots by simp
+  qed
+  then obtain i :: nat where "\<forall>\<P> \<in> carrier M. index_free \<P> (P, i)"
+    by blast
+
+  then have hyps:
\<comment> \<open>i\<close>   "field M"
\<comment> \<open>ii\<close>  "\<And>\<P>. \<P> \<in> carrier M \<Longrightarrow> carrier_coeff \<P>"
-    \<comment> \<open>iii\<close> "\<And>\<P>. \<P> \<in> carrier M \<Longrightarrow> index_free \<P> P"
+    \<comment> \<open>iii\<close> "\<And>\<P>. \<P> \<in> carrier M \<Longrightarrow> index_free \<P> (P, i)"
\<comment> \<open>iv\<close>  "\<zero>\<^bsub>M\<^esub> = indexed_const \<zero>"
-    using assms(1,3) no_roots unfolding extensions_def by auto
-  have degree_gt: "degree Q > 1"
-  proof (rule ccontr)
-    assume "\<not> degree Q > 1" hence "degree Q = 1"
-      using M.pirreducible_degree[OF M.carrier_is_subfield Q(1-2)] by simp
-    then obtain x where "x \<in> carrier M" and "M.eval Q x = \<zero>\<^bsub>M\<^esub>"
-      using M.degree_one_root[OF M.carrier_is_subfield Q(1)] M.add.inv_closed by blast
-    hence "M.eval (\<sigma> P) x = \<zero>\<^bsub>M\<^esub>"
-      using M.pdivides_imp_root_sharing[OF Q(1,3)] by simp
-    with \<open>x \<in> carrier M\<close> show False
-      using no_roots by simp
-  qed
+    using assms(1,3) unfolding extensions_def by auto

-  define image_poly where "image_poly = image_ring (eval_pmod M P Q) (poly_ring M)"
+  define image_poly where "image_poly = image_ring (eval_pmod M (P, i) Q) (poly_ring M)"
with \<open>degree Q > 1\<close> have "M \<lesssim> image_poly"
using image_poly_iso_incl[OF hyps Q(1)] by auto
moreover have is_field: "field image_poly"
@@ -530,7 +1745,7 @@
moreover have "image_poly \<in> extensions"
proof (auto simp add: extensions_def is_field)
fix \<P> assume "\<P> \<in> carrier image_poly"
-    then obtain R where \<P>: "\<P> = eval_pmod M P Q R" and "R \<in> carrier (poly_ring M)"
+    then obtain R where \<P>: "\<P> = eval_pmod M (P, i) Q R" and "R \<in> carrier (poly_ring M)"
unfolding image_poly_def image_ring_carrier by auto
hence "M.pmod R Q \<in> carrier (poly_ring M)"
using M.long_division_closed(2)[OF M.carrier_is_subfield _ Q(1)] by simp
@@ -545,32 +1760,32 @@
from \<open>M \<lesssim> image_poly\<close> interpret Id: ring_hom_ring M image_poly id
using iso_inclE[OF M.ring_axioms field.is_ring[OF is_field]] by simp

-    fix \<P> S
-    assume A: "\<P> \<in> carrier image_poly" "\<not> index_free \<P> S" "S \<in> carrier (poly_ring R)"
-    have "\<X>\<^bsub>S\<^esub> \<in> carrier image_poly \<and> Id.eval (\<sigma> S) \<X>\<^bsub>S\<^esub> = \<zero>\<^bsub>image_poly\<^esub>"
+    fix \<P> S j
+    assume A: "\<P> \<in> carrier image_poly" "\<not> index_free \<P> (S, j)" "S \<in> carrier (poly_ring R)"
+    have "\<X>\<^bsub>(S, j)\<^esub> \<in> carrier image_poly \<and> Id.eval (\<sigma> S) \<X>\<^bsub>(S, j)\<^esub> = \<zero>\<^bsub>image_poly\<^esub>"
proof (cases)
-      assume "P \<noteq> S"
-      then obtain Q' where "Q' \<in> carrier M" and "\<not> index_free Q' S"
+      assume "(P, i) \<noteq> (S, j)"
+      then obtain Q' where "Q' \<in> carrier M" and "\<not> index_free Q' (S, j)"
using A(1) image_poly_index_free[OF hyps Q(1) _ A(2)] unfolding image_poly_def by auto
-      hence "\<X>\<^bsub>S\<^esub> \<in> carrier M" and "M.eval (\<sigma> S) \<X>\<^bsub>S\<^esub> = \<zero>\<^bsub>M\<^esub>"
+      hence "\<X>\<^bsub>(S, j)\<^esub> \<in> carrier M" and "M.eval (\<sigma> S) \<X>\<^bsub>(S, j)\<^esub> = \<zero>\<^bsub>M\<^esub>"
using assms(1) A(3) unfolding extensions_def by auto
moreover have "\<sigma> S \<in> carrier (poly_ring M)"
using polynomial_hom[OF Hom.homh field_axioms M.field_axioms A(3)] unfolding \<sigma>_def .
ultimately show ?thesis
using Id.eval_hom[OF M.carrier_is_subring] Id.hom_closed Id.hom_zero by auto
next
-      assume "\<not> P \<noteq> S" hence S: "P = S"
+      assume "\<not> (P, i) \<noteq> (S, j)" hence S: "(P, i) = (S, j)"
by simp
have poly_hom: "R \<in> carrier (poly_ring image_poly)" if "R \<in> carrier (poly_ring M)" for R
using polynomial_hom[OF Id.homh M.field_axioms is_field that] by simp
-      have "\<X>\<^bsub>S\<^esub> \<in> carrier image_poly"
+      have "\<X>\<^bsub>(S, j)\<^esub> \<in> carrier image_poly"
using eval_pmod_var(2)[OF hyps Hom.homh Q(1) degree_gt] unfolding image_poly_def S by simp
-      moreover have "Id.eval Q \<X>\<^bsub>S\<^esub> = \<zero>\<^bsub>image_poly\<^esub>"
+      moreover have "Id.eval Q \<X>\<^bsub>(S, j)\<^esub> = \<zero>\<^bsub>image_poly\<^esub>"
using image_poly_eval_indexed_var[OF hyps Hom.homh Q(1) degree_gt Q(2)] unfolding image_poly_def S by simp
moreover have "Q pdivides\<^bsub>image_poly\<^esub> (\<sigma> S)"
proof -
obtain R where R: "R \<in> carrier (poly_ring M)" "\<sigma> S = Q \<otimes>\<^bsub>poly_ring M\<^esub> R"
-          using Q(3) unfolding S pdivides_def by auto
+          using Q(3) S unfolding pdivides_def by auto
moreover have "set Q \<subseteq> carrier M" and "set R \<subseteq> carrier M"
using Q(1) R(1) unfolding sym[OF univ_poly_carrier] polynomial_def by auto
ultimately have "Id.normalize (\<sigma> S) = Q \<otimes>\<^bsub>poly_ring image_poly\<^esub> R"
@@ -590,23 +1805,22 @@
ultimately show ?thesis
using domain.pdivides_imp_root_sharing[OF field.axioms(1)[OF is_field], of Q] by auto
qed
-    thus "\<X>\<^bsub>S\<^esub> \<in> carrier image_poly" and "Id.eval (\<sigma> S) \<X>\<^bsub>S\<^esub> = \<zero>\<^bsub>image_poly\<^esub>"
+    thus "\<X>\<^bsub>(S, j)\<^esub> \<in> carrier image_poly" and "Id.eval (\<sigma> S) \<X>\<^bsub>(S, j)\<^esub> = \<zero>\<^bsub>image_poly\<^esub>"
by auto
qed
ultimately have "law_restrict M = law_restrict image_poly"
using assms(2) by simp
hence "carrier M = carrier image_poly"
-  moreover have "\<X>\<^bsub>P\<^esub> \<in> carrier image_poly"
+  moreover have "\<X>\<^bsub>(P, i)\<^esub> \<in> carrier image_poly"
using eval_pmod_var(2)[OF hyps Hom.homh Q(1) degree_gt] unfolding image_poly_def by simp
-  moreover have "\<X>\<^bsub>P\<^esub> \<notin> carrier M"
-    using indexed_var_not_index_free[of P] hyps(3) by blast
+  moreover have "\<X>\<^bsub>(P, i)\<^esub> \<notin> carrier M"
+    using indexed_var_not_index_free[of "(P, i)"] hyps(3) by blast
ultimately show False by simp
qed

lemma (in field) exists_extension_with_roots:
-  shows "\<exists>L \<in> extensions. \<forall>P \<in> carrier (poly_ring R).
-    degree P > 0 \<longrightarrow> (\<exists>x \<in> carrier L. (ring.eval L) (\<sigma> P) x = \<zero>\<^bsub>L\<^esub>)"
+  shows "\<exists>L \<in> extensions. \<forall>P \<in> carrier (poly_ring R). (ring.splitted L) (\<sigma> P)"
proof -
obtain M where "M \<in> extensions" and "\<forall>L \<in> extensions. M \<lesssim> L \<longrightarrow> law_restrict L = law_restrict M"
using exists_maximal_extension iso_incl_hom by blast
@@ -619,17 +1833,16 @@

locale algebraic_closure = field L + subfield K L for L (structure) and K +
assumes algebraic_extension: "x \<in> carrier L \<Longrightarrow> (algebraic over K) x"
-    and roots_over_subfield: "\<lbrakk> P \<in> carrier (K[X]); degree P > 0 \<rbrakk> \<Longrightarrow> \<exists>x \<in> carrier L. eval P x = \<zero>\<^bsub>L\<^esub>"
+    and roots_over_subfield: "P \<in> carrier (K[X]) \<Longrightarrow> splitted P"

locale algebraically_closed = field L for L (structure) +
-  assumes roots_over_carrier: "\<lbrakk> P \<in> carrier (poly_ring L); degree P > 0 \<rbrakk> \<Longrightarrow> \<exists>x \<in> carrier L. eval P x = \<zero>\<^bsub>L\<^esub>"
+  assumes roots_over_carrier: "P \<in> carrier (poly_ring L) \<Longrightarrow> splitted P"

-definition (in field) closure :: "(('a list) multiset \<Rightarrow> 'a) ring" ("\<Omega>")
+definition (in field) closure :: "(('a list \<times> nat) multiset \<Rightarrow> 'a) ring" ("\<Omega>")
where "closure = (SOME L \<comment> \<open>such that\<close>.
-           \<comment> \<open>i\<close>  algebraic_closure L (indexed_const ` (carrier R)) \<and>
+           \<comment> \<open>i\<close>  algebraic_closure L (indexed_const ` (carrier R)) \<and>
\<comment> \<open>ii\<close> indexed_const \<in> ring_hom R L)"

-
lemma algebraic_hom:
assumes "h \<in> ring_hom R S" and "field R" and "field S" and "subfield K R" and "x \<in> carrier R"
shows "((ring.algebraic R) over K) x \<Longrightarrow> ((ring.algebraic S) over (h ` K)) (h x)"
@@ -648,12 +1861,11 @@
qed

lemma (in field) exists_closure:
-  obtains L :: "(('a list multiset) \<Rightarrow> 'a) ring"
+  obtains L :: "((('a list \<times> nat) multiset) \<Rightarrow> 'a) ring"
where "algebraic_closure L (indexed_const ` (carrier R))" and "indexed_const \<in> ring_hom R L"
proof -
obtain L where "L \<in> extensions"
-    and roots: "\<And>P. \<lbrakk> P \<in> carrier (poly_ring R); degree P > 0 \<rbrakk> \<Longrightarrow>
-                      \<exists>x \<in> carrier L. (ring.eval L) (\<sigma> P) x = \<zero>\<^bsub>L\<^esub>"
+    and roots: "\<And>P. P \<in> carrier (poly_ring R) \<Longrightarrow> (ring.splitted L) (\<sigma> P)"
using exists_extension_with_roots by auto

let ?K = "indexed_const ` (carrier R)"
@@ -685,7 +1897,7 @@
next
show "?K \<subseteq> carrier ?M"
proof
-      fix x :: "('a list multiset) \<Rightarrow> 'a"
+      fix x :: "(('a list \<times> nat) multiset) \<Rightarrow> 'a"
assume "x \<in> ?K"
hence "x \<in> carrier L"
using ring_hom_memE(1)[OF ring_hom_ring.homh[OF hom]] by auto
@@ -700,54 +1912,126 @@
proof (intro algebraic_closure.intro[OF M is_subfield])
have "(Id.R.algebraic over ?K) x" if "x \<in> carrier ?M" for x
using that Id.S.algebraic_consistent[OF subfieldE(1)[OF set_of_algs]] by simp
-    moreover have "\<exists>x \<in> carrier ?M. Id.R.eval P x = \<zero>\<^bsub>?M\<^esub>"
-      if "P \<in> carrier (?K[X]\<^bsub>?M\<^esub>)" and "degree P > 0" for P
+    moreover have "Id.R.splitted P" if "P \<in> carrier (?K[X]\<^bsub>?M\<^esub>)" for P
proof -
-      from \<open>P \<in> carrier (?K[X]\<^bsub>?M\<^esub>)\<close> have "P \<in> carrier (?K[X]\<^bsub>L\<^esub>)"
-        unfolding Id.S.univ_poly_consistent[OF subfieldE(1)[OF set_of_algs]] .
-      hence "set P \<subseteq> ?K"
-        unfolding sym[OF univ_poly_carrier] polynomial_def by auto
-      hence "\<exists>Q. set Q \<subseteq> carrier R \<and> P = \<sigma> Q"
-      proof (induct P, simp add: \<sigma>_def)
-        case (Cons p P)
-        then obtain q Q where "q \<in> carrier R" "set Q \<subseteq> carrier R" and "\<sigma> Q = P""indexed_const q = p"
-          unfolding \<sigma>_def by auto
-        hence "set (q # Q) \<subseteq> carrier R" and "\<sigma> (q # Q) = (p # P)"
-          unfolding \<sigma>_def by auto
-        thus ?case
-          by metis
-      qed
-      then obtain Q where "set Q \<subseteq> carrier R" and "\<sigma> Q = P"
-        by auto
-      moreover have "lead_coeff Q \<noteq> \<zero>"
-      proof (rule ccontr)
-        assume "\<not> lead_coeff Q \<noteq> \<zero>" then have "lead_coeff Q = \<zero>"
+      from \<open>P \<in> carrier (?K[X]\<^bsub>?M\<^esub>)\<close> have "P \<in> carrier (poly_ring ?M)"
+        using Id.R.carrier_polynomial_shell[OF subfieldE(1)[OF is_subfield]] by simp
+      show ?thesis
+      proof (cases "degree P = 0")
+        case True with \<open>P \<in> carrier (poly_ring ?M)\<close> show ?thesis
+          using domain.degree_zero_imp_splitted[OF field.axioms(1)[OF M]]
+          by fastforce
+      next
+        case False then have "degree P > 0"
by simp
-        with \<open>\<sigma> Q = P\<close> and \<open>degree P > 0\<close> have "lead_coeff P = indexed_const \<zero>"
-          unfolding \<sigma>_def by (metis diff_0_eq_0 length_map less_irrefl_nat list.map_sel(1) list.size(3))
-        hence "lead_coeff P = \<zero>\<^bsub>L\<^esub>"
-          using ring_hom_zero[OF ring_hom_ring.homh ring_hom_ring.axioms(1-2)] hom by auto
-        with \<open>degree P > 0\<close> have "\<not> P \<in> carrier (?K[X]\<^bsub>?M\<^esub>)"
+        from \<open>P \<in> carrier (?K[X]\<^bsub>?M\<^esub>)\<close> have "P \<in> carrier (?K[X]\<^bsub>L\<^esub>)"
+          unfolding Id.S.univ_poly_consistent[OF subfieldE(1)[OF set_of_algs]] .
+        hence "set P \<subseteq> ?K"
+          unfolding sym[OF univ_poly_carrier] polynomial_def by auto
+        hence "\<exists>Q. set Q \<subseteq> carrier R \<and> P = \<sigma> Q"
+        proof (induct P, simp add: \<sigma>_def)
+          case (Cons p P)
+          then obtain q Q where "q \<in> carrier R" "set Q \<subseteq> carrier R"
+            and "\<sigma> Q = P" "indexed_const q = p"
+            unfolding \<sigma>_def by auto
+          hence "set (q # Q) \<subseteq> carrier R" and "\<sigma> (q # Q) = (p # P)"
+            unfolding \<sigma>_def by auto
+          thus ?case
+            by metis
+        qed
+        then obtain Q where "set Q \<subseteq> carrier R" and "\<sigma> Q = P"
+          by auto
+        moreover have "lead_coeff Q \<noteq> \<zero>"
+        proof (rule ccontr)
+          assume "\<not> lead_coeff Q \<noteq> \<zero>" then have "lead_coeff Q = \<zero>"
+            by simp
+          with \<open>\<sigma> Q = P\<close> and \<open>degree P > 0\<close> have "lead_coeff P = indexed_const \<zero>"
+            unfolding \<sigma>_def by (metis diff_0_eq_0 length_map less_irrefl_nat list.map_sel(1) list.size(3))
+          hence "lead_coeff P = \<zero>\<^bsub>L\<^esub>"
+            using ring_hom_zero[OF ring_hom_ring.homh ring_hom_ring.axioms(1-2)] hom by auto
+          with \<open>degree P > 0\<close> have "\<not> P \<in> carrier (?K[X]\<^bsub>?M\<^esub>)"
+            unfolding sym[OF univ_poly_carrier] polynomial_def by auto
+          with \<open>P \<in> carrier (?K[X]\<^bsub>?M\<^esub>)\<close> show False
+            by simp
+        qed
+        ultimately have "Q \<in> carrier (poly_ring R)"
unfolding sym[OF univ_poly_carrier] polynomial_def by auto
-        with \<open>P \<in> carrier (?K[X]\<^bsub>?M\<^esub>)\<close> show False
-          by simp
+        with \<open>\<sigma> Q = P\<close> have "Id.S.splitted P"
+          using roots[of Q] by simp
+
+        from \<open>P \<in> carrier (poly_ring ?M)\<close> show ?thesis
+        proof (rule field.trivial_factors_imp_splitted[OF M])
+          fix R
+          assume R: "R \<in> carrier (poly_ring ?M)" "pirreducible\<^bsub>?M\<^esub> (carrier ?M) R" and "R pdivides\<^bsub>?M\<^esub> P"
+
+          from \<open>P \<in> carrier (poly_ring ?M)\<close> and \<open>R \<in> carrier (poly_ring ?M)\<close>
+          have "P \<in> carrier ((?set_of_algs)[X]\<^bsub>L\<^esub>)" and "R \<in> carrier ((?set_of_algs)[X]\<^bsub>L\<^esub>)"
+            unfolding Id.S.univ_poly_consistent[OF subfieldE(1)[OF set_of_algs]] by auto
+          hence in_carrier: "P \<in> carrier (poly_ring L)" "R \<in> carrier (poly_ring L)"
+            using Id.S.carrier_polynomial_shell[OF subfieldE(1)[OF set_of_algs]] by auto
+
+          from \<open>R pdivides\<^bsub>?M\<^esub> P\<close> have "R divides\<^bsub>((?set_of_algs)[X]\<^bsub>L\<^esub>)\<^esub> P"
+            unfolding pdivides_def Id.S.univ_poly_consistent[OF subfieldE(1)[OF set_of_algs]]
+            by simp
+          with \<open>P \<in> carrier ((?set_of_algs)[X]\<^bsub>L\<^esub>)\<close> and \<open>R \<in> carrier ((?set_of_algs)[X]\<^bsub>L\<^esub>)\<close>
+          have "R pdivides\<^bsub>L\<^esub> P"
+            using domain.pdivides_iff_shell[OF field.axioms(1)[OF L] set_of_algs, of R P] by simp
+          with \<open>Id.S.splitted P\<close> and \<open>degree P \<noteq> 0\<close> have "Id.S.splitted R"
+            using field.pdivides_imp_splitted[OF L in_carrier(2,1)] by fastforce
+          show "degree R \<le> 1"
+          proof (cases "Id.S.roots R = {#}")
+            case True with \<open>Id.S.splitted R\<close> show ?thesis
+              unfolding Id.S.splitted_def by simp
+          next
+            case False with \<open>R \<in> carrier (poly_ring L)\<close>
+            obtain a where "a \<in> carrier L" and "a \<in># Id.S.roots R"
+              and "[ \<one>\<^bsub>L\<^esub>, \<ominus>\<^bsub>L\<^esub> a ] \<in> carrier (poly_ring L)" and pdiv: "[ \<one>\<^bsub>L\<^esub>, \<ominus>\<^bsub>L\<^esub> a ] pdivides\<^bsub>L\<^esub> R"
+              using domain.not_empty_rootsE[OF field.axioms(1)[OF L], of R] by blast
+
+            from \<open>P \<in> carrier (?K[X]\<^bsub>L\<^esub>)\<close>
+            have "(Id.S.algebraic over ?K) a"
+            proof (rule Id.S.algebraicI)
+              from \<open>degree P \<noteq> 0\<close> show "P \<noteq> []"
+                by auto
+            next
+              from \<open>a \<in># Id.S.roots R\<close> and \<open>R \<in> carrier (poly_ring L)\<close>
+              have "Id.S.eval R a = \<zero>\<^bsub>L\<^esub>"
+                using domain.roots_mem_iff_is_root[OF field.axioms(1)[OF L]]
+                unfolding Id.S.is_root_def by auto
+              with \<open>R pdivides\<^bsub>L\<^esub> P\<close> and \<open>a \<in> carrier L\<close> show "Id.S.eval P a = \<zero>\<^bsub>L\<^esub>"
+                using domain.pdivides_imp_root_sharing[OF field.axioms(1)[OF L] in_carrier(2)] by simp
+            qed
+            with \<open>a \<in> carrier L\<close> have "a \<in> ?set_of_algs"
+              by simp
+            hence "[ \<one>\<^bsub>L\<^esub>, \<ominus>\<^bsub>L\<^esub> a ] \<in> carrier ((?set_of_algs)[X]\<^bsub>L\<^esub>)"
+              using subringE(3,5)[of ?set_of_algs L] subfieldE(1,6)[OF set_of_algs]
+              unfolding sym[OF univ_poly_carrier] polynomial_def by simp
+            hence "[ \<one>\<^bsub>L\<^esub>, \<ominus>\<^bsub>L\<^esub> a ] \<in> carrier (poly_ring ?M)"
+              unfolding Id.S.univ_poly_consistent[OF subfieldE(1)[OF set_of_algs]] by simp
+
+            from \<open>[ \<one>\<^bsub>L\<^esub>, \<ominus>\<^bsub>L\<^esub> a ] \<in> carrier ((?set_of_algs)[X]\<^bsub>L\<^esub>)\<close>
+             and \<open>R \<in> carrier ((?set_of_algs)[X]\<^bsub>L\<^esub>)\<close>
+            have "[ \<one>\<^bsub>L\<^esub>, \<ominus>\<^bsub>L\<^esub> a ] divides\<^bsub>(?set_of_algs)[X]\<^bsub>L\<^esub>\<^esub> R"
+              using pdiv domain.pdivides_iff_shell[OF field.axioms(1)[OF L] set_of_algs] by simp
+            hence "[ \<one>\<^bsub>L\<^esub>, \<ominus>\<^bsub>L\<^esub> a ] divides\<^bsub>poly_ring ?M\<^esub> R"
+              unfolding pdivides_def Id.S.univ_poly_consistent[OF subfieldE(1)[OF set_of_algs]]
+              by simp
+
+            have "[ \<one>\<^bsub>L\<^esub>, \<ominus>\<^bsub>L\<^esub> a ] \<notin> Units (poly_ring ?M)"
+              using Id.R.univ_poly_units[OF field.carrier_is_subfield[OF M]] by force
+            with \<open>[ \<one>\<^bsub>L\<^esub>, \<ominus>\<^bsub>L\<^esub> a ] \<in> carrier (poly_ring ?M)\<close> and \<open>R \<in> carrier (poly_ring ?M)\<close>
+             and \<open>[ \<one>\<^bsub>L\<^esub>, \<ominus>\<^bsub>L\<^esub> a ] divides\<^bsub>poly_ring ?M\<^esub> R\<close>
+            have "[ \<one>\<^bsub>L\<^esub>, \<ominus>\<^bsub>L\<^esub> a ] \<sim>\<^bsub>poly_ring ?M\<^esub> R"
+              using Id.R.divides_pirreducible_condition[OF R(2)] by auto
+            with \<open>[ \<one>\<^bsub>L\<^esub>, \<ominus>\<^bsub>L\<^esub> a ] \<in> carrier (poly_ring ?M)\<close> and \<open>R \<in> carrier (poly_ring ?M)\<close>
+            have "degree R = 1"
+              using domain.associated_polynomials_imp_same_length[OF field.axioms(1)[OF M]
+                    Id.R.carrier_is_subring, of "[ \<one>\<^bsub>L\<^esub>, \<ominus>\<^bsub>L\<^esub> a ]" R] by force
+            thus ?thesis
+              by simp
+          qed
+        qed
qed
-      ultimately have "Q \<in> carrier (poly_ring R)"
-        unfolding sym[OF univ_poly_carrier] polynomial_def by auto
-      moreover from \<open>degree P > 0\<close> and \<open>\<sigma> Q = P\<close> have "degree Q > 0"
-        unfolding \<sigma>_def by auto
-      ultimately obtain x where "x \<in> carrier L" and "Id.S.eval P x = \<zero>\<^bsub>L\<^esub>"
-        using roots[of Q] unfolding \<open>\<sigma> Q = P\<close> by auto
-      hence "Id.R.eval P x = \<zero>\<^bsub>?M\<^esub>"
-        unfolding Id.S.eval_consistent[OF subfieldE(1)[OF set_of_algs]] by simp
-      moreover from \<open>degree P > 0\<close> have "P \<noteq> []"
-        by auto
-      with \<open>P \<in> carrier (?K[X]\<^bsub>L\<^esub>)\<close> and \<open>Id.S.eval P x = \<zero>\<^bsub>L\<^esub>\<close> have "(Id.S.algebraic over ?K) x"
-        using Id.S.non_trivial_ker_imp_algebraic[of ?K x] unfolding a_kernel_def' by auto
-      with \<open>x \<in> carrier L\<close> have "x \<in> carrier ?M"
-        by auto
-      ultimately show ?thesis
-        by auto
qed
ultimately show "algebraic_closure_axioms ?M ?K"
unfolding algebraic_closure_axioms_def by auto
@@ -764,5 +2048,161 @@
using exists_closure unfolding closure_def
by (metis (mono_tags, lifting) someI2)+

+lemma (in field) algebraically_closedI:
+  assumes "\<And>p. \<lbrakk> p \<in> carrier (poly_ring R); degree p > 1 \<rbrakk> \<Longrightarrow> \<exists>x \<in> carrier R. eval p x = \<zero>"
+  shows "algebraically_closed R"
+proof
+  fix p assume "p \<in> carrier (poly_ring R)" thus "splitted p"
+  proof (induction "degree p" arbitrary: p rule: less_induct)
+    case less show ?case
+    proof (cases "degree p \<le> 1")
+      case True with \<open>p \<in> carrier (poly_ring R)\<close> show ?thesis
+        using degree_zero_imp_splitted degree_one_imp_splitted by fastforce
+    next
+      case False then have "degree p > 1"
+        by simp
+      with \<open>p \<in> carrier (poly_ring R)\<close> have "roots p \<noteq> {#}"
+        using assms[of p] roots_mem_iff_is_root[of p] unfolding is_root_def by force
+      then obtain a where a: "a \<in> carrier R" "a \<in># roots p"
+        and pdiv: "[ \<one>, \<ominus> a ] pdivides p" and in_carrier: "[ \<one>, \<ominus> a ] \<in> carrier (poly_ring R)"
+        using less(2) by blast
+      then obtain q where q: "q \<in> carrier (poly_ring R)" and p: "p = [ \<one>, \<ominus> a ] \<otimes>\<^bsub>poly_ring R\<^esub> q"
+        unfolding pdivides_def by blast
+      with \<open>degree p > 1\<close> have not_zero: "q \<noteq> []" and "p \<noteq> []"
+        using domain.integral_iff[OF univ_poly_is_domain[OF carrier_is_subring] in_carrier, of q]
+        by (auto simp add: univ_poly_zero[of R "carrier R"])
+      hence deg: "degree p = Suc (degree q)"
+        using poly_mult_degree_eq[OF carrier_is_subring] in_carrier q p
+        unfolding univ_poly_carrier sym[OF univ_poly_mult[of R "carrier R"]] by auto
+      hence "splitted q"
+        using less(1)[OF _ q] by simp
+      moreover have "roots p = add_mset a (roots q)"
+        using poly_mult_degree_one_monic_imp_same_roots[OF a(1) q not_zero] p by simp
+      ultimately show ?thesis
+        unfolding splitted_def deg by simp
+    qed
+  qed
+qed
+
+sublocale algebraic_closure \<subseteq> algebraically_closed
+proof (rule algebraically_closedI)
+  fix P assume in_carrier: "P \<in> carrier (poly_ring L)" and gt_one: "degree P > 1"
+  then have gt_zero: "degree P > 0"
+    by simp
+
+  define A where "A = finite_extension K P"
+
+  from \<open>P \<in> carrier (poly_ring L)\<close> have "set P \<subseteq> carrier L"
+    by (simp add: polynomial_incl univ_poly_carrier)
+  hence A: "subfield A L" and P: "P \<in> carrier (A[X])"
+    using finite_extension_mem[OF subfieldE(1)[OF subfield_axioms], of P] in_carrier
+          algebraic_extension finite_extension_is_subfield[OF subfield_axioms, of P]
+    unfolding sym[OF A_def] sym[OF univ_poly_carrier] polynomial_def by auto
+  from \<open>set P \<subseteq> carrier L\<close> have incl: "K \<subseteq> A"
+    using finite_extension_incl[OF subfieldE(3)[OF subfield_axioms]] unfolding A_def by simp
+
+  interpret UP_K: domain "K[X]"
+    using univ_poly_is_domain[OF subfieldE(1)[OF subfield_axioms]] .
+  interpret UP_A: domain "A[X]"
+    using univ_poly_is_domain[OF subfieldE(1)[OF A]] .
+  interpret Rupt: ring "Rupt A P"
+    unfolding rupture_def using ideal.quotient_is_ring[OF UP_A.cgenideal_ideal[OF P]] .
+  interpret Hom: ring_hom_ring "L \<lparr> carrier := A \<rparr>" "Rupt A P" "rupture_surj A P \<circ> poly_of_const"
+    using ring_hom_ringI2[OF subring_is_ring[OF subfieldE(1)] Rupt.ring_axioms
+          rupture_surj_norm_is_hom[OF subfieldE(1) P]] A by simp
+  let ?h = "rupture_surj A P \<circ> poly_of_const"
+
+  have h_simp: "rupture_surj A P ` poly_of_const ` E = ?h ` E" for E
+    by auto
+  hence aux_lemmas:
+    "subfield (rupture_surj A P ` poly_of_const ` K) (Rupt A P)"
+    "subfield (rupture_surj A P ` poly_of_const ` A) (Rupt A P)"
+    using Hom.img_is_subfield(2)[OF _ rupture_one_not_zero[OF A P gt_zero]]
+          ring.subfield_iff(1)[OF subring_is_ring[OF subfieldE(1)[OF A]]]
+          subfield_iff(2)[OF subfield_axioms] subfield_iff(2)[OF A] incl
+    by auto
+
+  have "carrier (K[X]) \<subseteq> carrier (A[X])"
+    using subsetI[of "carrier (K[X])" "carrier (A[X])"] incl
+    unfolding sym[OF univ_poly_carrier] polynomial_def by auto
+  hence "id \<in> ring_hom (K[X]) (A[X])"
+  hence "rupture_surj A P \<in> ring_hom (K[X]) (Rupt A P)"
+    using ring_hom_trans[OF _ rupture_surj_hom(1)[OF subfieldE(1)[OF A] P], of id] by simp
+  then interpret Hom': ring_hom_ring "K[X]" "Rupt A P" "rupture_surj A P"
+    using ring_hom_ringI2[OF UP_K.ring_axioms Rupt.ring_axioms] by simp
+
+  from \<open>id \<in> ring_hom (K[X]) (A[X])\<close> have Id: "ring_hom_ring (K[X]) (A[X]) id"
+    using ring_hom_ringI2[OF UP_K.ring_axioms UP_A.ring_axioms] by simp
+  hence "subalgebra (poly_of_const ` K) (carrier (K[X])) (A[X])"
+    using ring_hom_ring.img_is_subalgebra[OF Id _ UP_K.carrier_is_subalgebra[OF subfieldE(3)]]
+          univ_poly_subfield_of_consts[OF subfield_axioms] by auto
+
+  moreover from \<open>carrier (K[X]) \<subseteq> carrier (A[X])\<close> have "poly_of_const ` K \<subseteq> carrier (A[X])"
+    using subfieldE(3)[OF univ_poly_subfield_of_consts[OF subfield_axioms]] by simp
+
+  ultimately
+  have "subalgebra (rupture_surj A P ` poly_of_const ` K) (rupture_surj A P ` carrier (K[X])) (Rupt A P)"
+    using ring_hom_ring.img_is_subalgebra[OF rupture_surj_hom(2)[OF subfieldE(1)[OF A] P]] by simp
+
+  moreover have "Rupt.finite_dimension (rupture_surj A P ` poly_of_const ` K) (carrier (Rupt A P))"
+  proof (intro Rupt.telescopic_base_dim(1)[where
+            ?K = "rupture_surj A P ` poly_of_const ` K" and
+            ?F = "rupture_surj A P ` poly_of_const ` A" and
+            ?E = "carrier (Rupt A P)", OF aux_lemmas])
+    show "Rupt.finite_dimension (rupture_surj A P ` poly_of_const ` A) (carrier (Rupt A P))"
+      using Rupt.finite_dimensionI[OF rupture_dimension[OF A P gt_zero]] .
+  next
+    let ?h = "rupture_surj A P \<circ> poly_of_const"
+
+    from \<open>set P \<subseteq> carrier L\<close> have "finite_dimension K A"
+      using finite_extension_finite_dimension(1)[OF subfield_axioms, of P] algebraic_extension
+      unfolding A_def by auto
+    then obtain Us where Us: "set Us \<subseteq> carrier L" "A = Span K Us"
+      using exists_base subfield_axioms by blast
+    hence "?h ` A = Rupt.Span (?h ` K) (map ?h Us)"
+      using Hom.Span_hom[of K Us] incl Span_base_incl[OF subfield_axioms, of Us]
+      unfolding Span_consistent[OF subfieldE(1)[OF A]] by simp
+    moreover have "set (map ?h Us) \<subseteq> carrier (Rupt A P)"
+      using Span_base_incl[OF subfield_axioms Us(1)] ring_hom_memE(1)[OF Hom.homh]
+      unfolding sym[OF Us(2)] by auto
+    ultimately
+    show "Rupt.finite_dimension (rupture_surj A P ` poly_of_const ` K) (rupture_surj A P ` poly_of_const ` A)"
+      using Rupt.Span_finite_dimension[OF aux_lemmas(1)] unfolding h_simp by simp
+  qed
+
+  moreover have "rupture_surj A P ` carrier (A[X]) = carrier (Rupt A P)"
+    unfolding rupture_def FactRing_def A_RCOSETS_def' by auto
+  with \<open>carrier (K[X]) \<subseteq> carrier (A[X])\<close> have "rupture_surj A P ` carrier (K[X]) \<subseteq> carrier (Rupt A P)"
+    by auto
+
+  ultimately
+  have "Rupt.finite_dimension (rupture_surj A P ` poly_of_const ` K) (rupture_surj A P ` carrier (K[X]))"
+    using Rupt.subalbegra_incl_imp_finite_dimension[OF aux_lemmas(1)] by simp
+
+  hence "\<not> inj_on (rupture_surj A P) (carrier (K[X]))"
+    using Hom'.infinite_dimension_hom[OF _ rupture_one_not_zero[OF A P gt_zero] _
+          UP_K.carrier_is_subalgebra[OF subfieldE(3)] univ_poly_infinite_dimension[OF subfield_axioms]]
+          univ_poly_subfield_of_consts[OF subfield_axioms]
+    by auto
+  then obtain Q where Q: "Q \<in> carrier (K[X])" "Q \<noteq> []" and "rupture_surj A P Q = \<zero>\<^bsub>Rupt A P\<^esub>"
+    using Hom'.trivial_ker_imp_inj Hom'.hom_zero unfolding a_kernel_def' univ_poly_zero by blast
+  with \<open>carrier (K[X]) \<subseteq> carrier (A[X])\<close> have "Q \<in> PIdl\<^bsub>A[X]\<^esub> P"
+    using ideal.rcos_const_imp_mem[OF UP_A.cgenideal_ideal[OF P]]
+    unfolding rupture_def FactRing_def by auto
+  then obtain R where "R \<in> carrier (A[X])" and "Q = R \<otimes>\<^bsub>A[X]\<^esub> P"
+    unfolding cgenideal_def by blast
+  with \<open>P \<in> carrier (A[X])\<close> have "P pdivides Q"
+    using dividesI[of _ "A[X]"] UP_A.m_comm pdivides_iff_shell[OF A] by simp
+  hence "splitted P"
+    using pdivides_imp_splitted[OF in_carrier
+          carrier_polynomial_shell[OF subfieldE(1)[OF subfield_axioms] Q(1)] Q(2)
+          roots_over_subfield[OF Q(1)]] Q by simp
+  with \<open>degree P > 1\<close> obtain a where "a \<in># roots P"
+    unfolding splitted_def by force
+  thus "\<exists>x\<in>carrier L. eval P x = \<zero>"
+    unfolding roots_mem_iff_is_root[OF in_carrier] is_root_def by blast
+qed
+
end
-
+```