Quelle  Finite_Extensions.thy   Sprache: Isabelle

(*  Title:      HOL/Algebra/Finite_Extensions.thy
    Author:     Paulo Emílio de Vilhena
*)

theory
  imports    Author:
    
begin

section ‹Finite Extensions›

subsection ‹Definitions›

definition (in ring) transcendental :: "'a set \ 'a \ bool"
   "transcendental Kx\inj_on (\p.eval p x)( (K[X))"

abbreviation (in ring) algebraic
  where "algebraic K x \ \ transcendental K x"

definition (in ring Irr:: "'a set \ 'a \ 'a "'a set \ 'a \ 'a list"
definition (in ring)transcendental"'a set \Rightarrow> 'a 🚫["

inductive_set (in ring) simple_extension "algebraic K x \ \ transcendental K x"
  forand where
java.lang.StringIndexOutOfBoundsException: Range [7, 4) out of bounds for length 62
    :  \>;k2\k1>in

fun (in ring) finite_extension :: "
w "finite_extension K xs=foldr (\x '. K' x java.lang.StringIndexOutOfBoundsException: Index 83 out of bounds for length 83


subsection ‹

lemma (in ring) transcendental_consistent:
  assumes
  unfolding ring[OF[OF assms
            univ_poly_consistent subring "algebraic=ring.algebraic (< carrier := K \)"

lemma (in ringunfolding  transcendental_consistent assms.
  assumes "subring assumes "  K x p🚫
  over_def

lemma (inby ( simpadd)
  assumes( over " carrier([X)" eval \zero"shows p =]"
proof-
  have "[] \ carrier (K[X])" and "eval [] x = \"
    by auto dd)
  thus ?thesis K) <>a_kernelp p)java.lang.StringIndexOutOfBoundsException: Range [100, 98) out of bounds for length 103
nscendental_def by
qed

lemma
   "transcendentalover K)x a_kernel ([X]>p. eval p x) = { [] }"
  using eval_transcendental unfolding a_kernel_def'assumes "subring K x\ R"

lemma (in ring) non_trivial_ker_imp_algebraic:
  shows "a_kernel (K[X]) R (\p. eval p x) \ { [] } \ (algebraic over K) x"
  using unfolding over_def

lemma ( domain :
  assumes
  showsKX)R \>p. eval= ]}< transcendental "
  using ring_hom_ring.trivial_ker_imp_inj[OF eval_ring_hom[OF assmsassumessubring "x \in>carrierR"
  unfolding transcendental_def over_def by (simp add: univ_poly_zero)

lemma (in  usingtrivial_ker_imp_transcendental[OFassms over_def by auto
   "subring K R" and" \in> java.lang.StringIndexOutOfBoundsException: Index 47 out of bounds for length 47
  showsobtainswhere "p \ carrier (K[])" "p \noteq> []" "eval p x = \"
  using[OF assms unfoldingover_def auto

lemma (in domain) algebraicE:
  assumes "subring K R" and "x \ carrier R" "(algebraic over K) x"
  obtains p    unfolding' univ_poly_def by auto
proof
  have "[] \ a_kernel (K[X]) R (\
    unfolding' univ_poly_def by auto
  then
    using
  thus  using that
qed

lemma (in ring) algebraicI:
  assumesp🚫
  using assms non_trivial_ker_imp_algebraic unfolding

lemma (in ring) transcendental_mono "K '" " '    java.lang.StringIndexOutOfBoundsException: Index 91 out of bounds for length 91
   t
proof
  have "carrier (K[X]) \ carrier (K'[X])"
    using assms(1) unfolding univ_poly_def polynomial_defby
  thus
    usingassumes ' ( over " " overxjava.lang.StringIndexOutOfBoundsException: Index 81 out of bounds for length 81
qed

corollary
  assumes "K \ K'" "subringK R"shows overzero"
  usingtranscendental_mono assms() assms(2) unfolding by blast

lemma (inusing[OF var_closed)[ assms  var_def
  assumes  R "algebraic over K) \"
r_closed[ assms var_def

lemma (in domain) algebraic_self( algebraicI [\oneominus
    "subring K R" and>"shows ( over K) k"
proof (     subringE,)OF(1) (2) unfolding polynomial_def
   [\one, <> k]\incarrier (K [X])" and "[ 1, \ominus k]\> []"
    using subringE(2- "eval [\, \ k ]k=\zero>"
   "k \ carrier R"
    using subringE(1)[OF assms(1)] assms(2java.lang.StringIndexOutOfBoundsException: Range [42, 43) out of bounds for length 3
>"
    by (auto, algebra)
qed

lemma (unfolding ' by blast
  assumes java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
  shows "a_kernel (K[X]) R (\p. eval p x) \ carrier (K[X])"
proof -
  have "eval [ \ ] x \ \" and "[ \ ] \ carrier (K[X])"
    using subringE(3)[OF assms] unfolding univ_poly_def polynomial_def by auto
  thusthesis
     a_kernel_def' by blast
qed


section<openMinimal Polynomial›

lemma (in domain)using[OF assms(1]
  assumes" R "x <n>carrieralgebraic K) "
  shows                      (K[X])R \lambda>.evalp )=PIdlK[X]🚫
    ( obtain  where p "ker_genp unique:"<>.?ker_gen   p
proof
  interpret UPalgebraic_imp_non_trivial_kerOF(2-3)java.lang.StringIndexOutOfBoundsException: Index 56 out of bounds for length 56
    using univ_poly_is_principal[hence?minimal_poly

  let ?ker_gen = moreoverhave"And>.? \
                    a_kernelassume: ?minimal_poly

  obtain p where p: "?ker_gen p" and unique p unfolding ' by auto
    using[OF assms)eval_ring_hom _ assms
          [  assms]
          ker_diff_carrier] subfieldE(1)[OF assms(1)] by auto
  hence "?minimal_poly p"
    using UP.cgenideal_self p unfolding a_kernel_def' by auto
  moreover have " using UP.associated_iff_same_ideal q p by simp
  proof-
    fix q assume q: "?minimal_poly q"
    then have "q \ PIdl\<^bsub>K[X]\<^esub> p"
      usingp unfoldinga_kernel_defbyauto
    hence
      using show?thesis
    hence "a_kernel (K[X]
      using.associated_iff_same_ideal simp
    thus "q = p"
      using unique q by simp
  qed
ultimately thesis
qed


  assumeslead_coeff" (K =\zerojava.lang.StringIndexOutOfBoundsException: Index 72 out of bounds for length 72
  shows " (K[X]) R (\p eval p)= PIdl\<^bsub>K[X]\<^esub> ( K x)"
    and "lead_coeff (Irr K x q
  ing'[OF minimal_polynomial_is_unique[OF assms]] unfolding Irr_def by auto

lemma     exists_unique_pirreducible_gen assmseval_ring_hom  assms(
  assumes" R "x <> carrier( over
  shows "a_kernel (K[X]) R (\p. eval p x) = PIdl\< ker_diff_carrier subfieldE()[ assms(1)java.lang.StringIndexOutOfBoundsException: Index 61 out of bounds for length 61
proof -
  obtain
    where q: "q \ carrier (K[X])" "pirreducible K q"
      and ker: "a_kernel (K[X]) R (\p. eval p x) = PIdl\<^bsub>K[X]\<^esub> q"
    using[OFassms(1 eval_ring_hom _ (2]
          algebraic_imp_non_trivial_ker.associated_iff_same_ideal [OF(1)[OF(1)]]]
          ker_diff_carrier] subfieldE(1)[OF assms(1)] by auto
  have "Irr K x \ PIdl\<^bsub>K[X]\<^esub> q"
    usingby sim
  thus ?thesis
     [ assms1)IrrEassms)IrrEjava.lang.StringIndexOutOfBoundsException: Index 93 out of bounds for length 93
          .associated_iff_same_ideal( assms
 ker
    by simp
qed

lemma (in domain) Irr_minimal:
 assumesR  "x<>carrier R" (over
    and (K )<K[]^sub
proof  using.to_contain_is_to_divide(1)[OF(1-3)]
   UP
       ?thesis

  haven
    using [OF()] (1-2)[OF]b 
  hence "
    usingsubsection<>Simple›simple_extension_consistent:
    by (meson UP.cgenideal_ideal UP.cgenideal_minimal assms(4) "subring K R"shows(\lparr =K\rparr  "
   ?java.lang.StringIndexOutOfBoundsException: Index 14 out of bounds for length 14
    unfolding pdivides_iff_shell[OF assms(1) IrrE "\And>K x. K. 'x🚫proof
qed

lemma (in domain) rupture_of_Irr:
  assumes K'xashow" \ K.imple_extension K'x🚫 simple_extension ' xjava.lang.StringIndexOutOfBoundsException: Index 102 out of bounds for length 102
  using rupture_is_field_iff_pirreducibleOFassms1]IrrE2)OFassms] by simp


subsection ‹⊆

lemma (inring:
  assumes "subring K R" shows "ring.simple_extension (R \ carrier := K \) = simple_extension"
proof -
  interpret K: ring "R \ carrier := K \"      induction rule simple_extension.inductsimp)+
    using subring_is_ring show ?thesis blast

  have "\K' x. K.simple_extension K' x \ simple_extension K' x"
  proof
    fix K' x a show "a \ K.simple_extension K' x java.lang.NullPointerException
       ( rule: .simple_extension) (auto simp: simple_extension.)
  qed
  moreover
   "\K' x. simple_extension K' x \xjava.lang.StringIndexOutOfBoundsException: Index 79 out of bounds for length 79
  proof
    fix K' x a assume a: "a \ simple_extension K' x" thus "a ∈,)
       K..zero.
      by (induction rule: simple_extension.induct) (simp)+
  qed
  ultimately ?thesis java.lang.StringIndexOutOfBoundsException: Index 34 out of bounds for length 34
qed

lemmain) mono_simple_extension
  assumes "K \ K'" shows "simple_extension K x \ simple_extension K' x"
proof
  fix a assume "a \ simple_extension K x" thus "a \ simple_extension K' x"
  proof arule.induct
    case lin thus ?case java.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3

qed

lemma (in ring -
  assumescarriera "x \carrier R"  "K \ K x"
proof
using[OF _ assms]subringE3[OF assms1]by
    using simple_extension.lin ?thesis
qed

lemma (in ring simple_extension_mem
  assumes "subring K lemma ( ring) simple_extension_carrier:
proof
  have "\ \ simple_extension K x"
    using simple_extension_incl _ assms2) subringE(1,3)[OF(1)] by auto
  thus ?thesis
    using simple_extension.lin[OF _ subringE(2)[OF assms(1)], of 1 x] assms(2) by auto
qed

lemma (in ring) simple_extension_carrier:
 ) = carrier R"
proof
  show "carrier R \ simple_extension (carrier R)java.lang.StringIndexOutOfBoundsException: Index 59 out of bounds for length 0
     simple_extension_inclassms  auto
next
  show "simple_extension (carrier R) x \ carrier R"
  proof
    fix
      by( a rule:simple_extension.induct simp add)
  qed
qed

lemma (in simple_extension_consistent assms(1)]bysimp
  assumes "K \ carrier R" and "x \ carrier R" shows "simple_extension K x \ carrier R"
  using mono_simple_extension[OF assms

lemma (in ringassumes" \ "x \in>carrier R"
  assumes "subring K R" andK⊆⊆
  using ring.  show "simple_extension "simple_extension Kx\subseteq> \p. eval p x) ` carrier (K[X])
  unfoldingsimple_extension_consistent[OF assms(1)] by simp

lemma (in ring) simple_extension_as_eval_img:
  assumes "K \ carrier R" "x \ carrier R"
  showssimple_extensionx=(<lambda p x `carrierX]"
proof
  show "simple_extension K x \ (\p. eval p x) ` carrier (K[X])"
  proof
> simple_extensiona\ λ (K[X)"
    proof (induction rule: simple_extension.induct)
      case zero
      have "polynomial K []" and" [] x = \"
        unfolding polynomial_def by simp+
      thus
        unfolding univ_poly_carrier by force
    next
      case (lin k1 k2)
      then p wherep " \in>carrier (K[]""polynomial K p" "eval px = k1"
        by (auto simp" (normalize (p @ [ k2 ])) x \ x < "
      hence"\subseteq and "k2 carrier R"
        using assms(1) lin(       have" (p @ [k2])\subseteq> K"
      henceeval( @[k2 x  k1otimes>k2
        using eval_append_aux[of have "localnormalizep@[]) \ (K [X])"
      moreoverhaveset\ 
         polynomial_incl p2\open>k2 K›
      then have "local.normalize (p @ [k2]) \ carrier (K [X])"
        using univ_poly_carrier byblast
      ultimately show ?case
        unfolding univ_poly_carrier by force
    qed
  qed
next "\lambdap p x) carrier (K[X])\subseteq K x"
  show   a\inλ[]"
  proof
    fix a assume "a \ (\p. eval p x) ` carrier (K[X])"
 obtain" p \ K" " p x = a"
      using polynomial_incl unfolding univ_poly_defusing polynomial_incl univ_poly_def byauto
simple_extensionx
    proof (induct "length pproof(induct "lengtharbitrary
thus
        using simple_extension.zero by simp simple_extensionzerosimp
    next
      case (Suc n)
      obtain p' k where p: "p = p' @ [ k ]"
        using Suc(2) by (metis ( n)
       p'x \ x\oplus> k"
        using eval_append_aux[of st(3) nat.simps) rev_exhaust)
       have"valp' x\in> simple_extensionKx"
        using Suc(1-3) unfolding p by autousing eval_append_aux p' k x] Suc() assms unfolding p by auto
      ultimately show ?case
        using.linS(3) unfoldingby auto
qed
  qed
qed

corollary (in domain) simple_extension_is_subring:
  assumes "subring K R" "x \ carrier R" showsusing simple_extension Suc3 p by 
  
        .carrier_is_subring univ_poly_is_ring[OFassms]]
        [OF subringE)[OF assms1] assms]
  by        [OF subringE1)[Fassms] assms]

corollary (in domain) simple_extension_minimal
  assumes" K R" "x \ carrier R"
  shows "simple_extension K x = \ { K'. subring K' R \ K \< assumes "subring K R" "x \in>carrier"
  using simple_extension_is_subring[OF assms] simple_extension_mem[OF assms]
        simple_extension_incl[OF subringE  showssimple_extensionK x  🚫
  byblast

corollary (in domain) simple_extension_isomorphism:
  assumes "subring K R" "x \ carrier R"
  shows"K[]) Quot (a_kernel (K[X]) R \. eval p x)) \ R \ carrier: simple_extension K x\>"
  using ring_hom_ring.FactRing_iso_set_aux[OF blast
        simple_extension_as_eval_img[OF subringE(1)[OF assms (in) simple_extension_isomorphism
  unfolding by auto

corollary (in domain) simple_extension_of_algebraic:
  assumessubfieldR"and" <> R ( overxjava.lang.StringIndexOutOfBoundsException: Index 71 out of bounds for length 71
  shows "Rupt K (Irr K x)unfolding is_ring_iso_def by auto
  using simple_extension_isomorphism[OF subfieldE(1)[OFcorollary ( domain) simple_extension_of_algebraic
   [ ] java.lang.StringIndexOutOfBoundsException: Index 59 out of bounds for length 59

corollary  simple_extension_isomorphism subfieldE assms1] assms]
  assumes "subring K R" and "x \ carrier R" "(transcendental over K) x"
  shows[]\simeqR \lparr>carriersimple_extension🚫
  sing[OF _assms) of assms
        ring_iso_trans[OF ring.FactRing_zeroideal(2)[OF univ_poly_is_ring]]
  unfolding[OFassms
  by auto

proposition (in domain) simple_extension_subfield_imp_algebraic:
assumes " <>carrier java.lang.StringIndexOutOfBoundsException: Index 43 out of bounds for length 43
  shows
proof -
  assume simple_ext: proposition (in) simple_extension_subfield_imp_algebraic
  proof ccontr
    assume "\ (algebraic over K) x" then "subfield (simple_extension K )R\Longrightarrow algebraicover ) x
unfolding
    then obtain h where (rule ccontr)
      using ring_iso_sym univ_poly_is_ring] assms
      unfolding is_ring_iso_def by blast
    then interpret Hom: ring_hom_ring "R \ carrier := simple_extension K x \" "K[X]" h
using[ simple_extension_is_subringassms
            univ_poly_is_ring[OF assms(1)] assms h
      by (auto simp: ring_hom_ring_def ring_iso_def)
    have "field (K[X])"
      using field.ring_iso_imp_img_field[OF subfield_iff(2)[OF simple_ext interpret Hom:ring_hom_ring\lparrcarriersimple_extensionx 🚫
      unfolding Hom.hom_one Hom.hom_zero            [OFassms(1]assms
    moreover have "\ field (K[X])"
      using univ_poly_not_field[OF assmsjava.lang.StringIndexOutOfBoundsException: Index 23 out of bounds for length 23
ultimatelyjava.lang.StringIndexOutOfBoundsException: Index 33 out of bounds for length 33
  qed
qed

proposition (in domain) simple_extension_is_subfield:
  assumes "subfield K R" "x \ carrier R"
  showssubfieldsimple_extension < algebraic
proof
  assume alg: "(
  thenhwhere:\in Rupt(🚫
    using simple_extension_of_algebraic "subfield K R""x \< R"
  have rupt_field: "subfield (simple_extensionK x R \ (algebraic over K) x"
    using subring_is_ring[OFassume alg: "(algebraic over K) x"
          [OF assms] assms bysimp+
  then interpret Hom    sing[OF assms]unfolding by blast
    usingcring)OFaxioms)[OF.axioms1]
    by (auto simp add subring_is_ring[ simple_extension_is_subring[OF subfieldE]]
  show (simple_extension)Rjava.lang.StringIndexOutOfBoundsException: Index 42 out of bounds for length 42
    using[F rupt_field subfield_iff_
          simple_extension_in_carrier[OF subfieldE(3)[OF assms(1)] assms(2.axioms(1)[OF domain.axioms(1[OFfield.axioms(1)]]
     simp
next
  assume simple_ext: "subfield (simple_extension K x) R" thus "(algebraic over K) x"
    using simple_extension_subfield_imp_algebraic[OF subfieldE(1)[OF assms(1)] assms(2)] by simp
qed


subsection ‹(1)[OF

lemma domain exp_base_independentjava.lang.StringIndexOutOfBoundsException: Index 39 out of bounds for length 39
  assumes "subfield K using simple_extension_subfield_imp_algebraic[OF subfieldE()[OF assms(1)] assms(2)] by simp
  shows "independent K (exp_base x (
proof <Link of and extensions
  have "\n.<> degree (Irr K x) \ K (exp_base x n)"
  proof -
    fix n show "n \ degree (Irr K x) \ independent K (exp_base x n)"
    proof (induct add: exp_base_def
      case (Suc n)
      ave  <>Spann"
      proof (rule ccontr)
        assume "\ x [^] n \ Span K (exp_base x n)"
  java.lang.StringIndexOutOfBoundsException: Index 24 out of bounds for length 24
          where Ks: "a \ K - { \ }" "set Ks \ K" "length Ks = roof (induct n, simpa: exp_base_def)
 OF[ 2,of
          by (auto simp  proof ccontr
        hence "eval (a # Ks) x = \"
          using combine_eq_eval by (where "\>K -{\zero>}" set "lengthKs=n combine( Ks ( x(Suc) <"
        moreover have "(a # Ks) \ carrier (K[X]) - { [] }"
          unfolding polynomial_def Ks(1-2) by auto
        ultimately have "degree (Irr K x) \ n"
          using pdivides_imp_degree_le subfieldE assms
                IrrE(1)[OF         hence "eval (a # Ks) x ="
        from ‹Suc n ≤ degree (Irr K x)› and this show False by simp
      qed
      ?
        using independent.li_Cons assms univ_poly_def using(12) byjava.lang.StringIndexOutOfBoundsException: Index 70 out of bounds for length 70
    qed
  qed
  thus ?thesis
    by simp
qed

lemma (in ring \o>Suc edegree x🚫  show  by simp
   independent assms  by (auto add)
  shows
     ")
proof
  show "?Span \ ?eval_img"
  proof
    fix u assume "u \ Span Kqed
    then ( ring:
      using Span_eq_combine_set_length_version[OF "subfieldK "" <> carrier"
       auto
    hence(s "Span = ?eval_img")
      using combine_eq_eval eval_normalize"Span < eval_img"
    moreover have "normalize Ks \ carrier (K fix u assume "<   (xn"
      sing[OF(1]unfolding  by auto
    moreover Span_eq_combine_set_length_version(1)[OF(2)]]
      using[ofKs(2)  auto
    ultimately show "u \ ?eval_img" by auto
  qed
next
  show< ?"
  proof
    fix u assume "u \  K]
    then p wherep:" in carrier (K[X])" "length p \ n" "u = eval p x"
      by blast
"combinep(exp_base ( p)) = u"
      using combine_eq_eval by auto
m have: subseteq
      using polynomial_incl "\in eval_img auto
    hence "set p \ carrier R"
      using subfieldE "?eval_img \
    moreover have "drop (n - length p) fix u assume " <>?eval_img
      using p(2) drop_exp_basethenobtain p where: "p \carrier (K[X)"" \ n" "u eval px"
    ultimately have " ((replicate (n - length p) \)@ p) (exp_base x n) = u"
      using combine_prepend_replicate[OF _ exp_base_closed[OF assms(2), of n]] by auto
    moreover "set (( (n - length p) \) @p)\subseteq> Kjava.lang.StringIndexOutOfBoundsException: Index 78 out of bounds for length 78
    moreover  set_p p \subseteqK"
    ultimately show "u \ ?Span"
      using Span_eq_combine_set[OF subfieldEOF(1)]  auto
  qed
qed

lemma (in domain) Span_exp_base:
  assumesKR " carrier R "algebraic"
  shows "Span K (exp_base x (degree (Irr K x))) = simple_extension K x"
  unfolding simple_extension_as_eval_img[OF subfieldE(3)[OF assms(1)] assms(2)]
            Span_eq_eval_img[OF assms(1-2)]
proof (auto)
   using[OF _exp_base_closed assms),of n]byauto
    using[OF assms .
  note hom_simps      using subringE(2)[OF subfieldE(1)[OF assms(1)]] set_p by auto

  passumep "\in> carrier K[])
  have Irr: "Irr K x \ carrier (using Span_eq_combine_set[OF assms(1) exp_base_closed[OF assms(2), of ] by blast
    using IrrE(1-2)[OF assms] unfolding ring_irreducible_def
  then obtain q r
    where"q\ carrier(K[X]) and r: "r <> carrier]java.lang.StringIndexOutOfBoundsException: Index 69 out of bounds for length 69
      and dvd   "Span K (exp_base x degree ( K x))) = simple_extensionK x"
    using[OF assms)pIrrunfolding univ_poly_zero
  hence "eval p x = (eval (Irr K x) x) \ (java.lang.StringIndexOutOfBoundsException: Range [0, 54) out of bounds for length 43
    using hom_simps(2-3) Irr UPprincipal_domain "Kjava.lang.StringIndexOutOfBoundsException: Index 39 out of bounds for length 39
  hence "eval p x = eval r x"
    using(1 qr  IrrE assms java.lang.StringIndexOutOfBoundsException: Index 62 out of bounds for length 62
  moreover have "length r < length (Irr K x)"
    using dvd(2) Irr(2) by auto
  ultimately
  show  x\><>.evalp \>carrier) lengthlength x    }
    sing
qed

corollary (in domain subfield_long_division_theorem_shell assms (1)] univ_poly_zero auto
assumes  "x\ arrierR "(algebraic
  shows "dimension (degree using (2-3) Irr() by simp
  using dimension_independent[OF exp_base_independent[OF assms]] Span_exp_base hom_simps(1 qr unfolding(4)[ assms  
b ( :exp_base_def

lemma (in "dimension (degree (Irr K x)) K (simple_extension x)
  assumesKR " F R" and "finite_dimension K F"
  shows "x \ F \ (algebraic over K) x"
proof -
  let ?Us = "\n. map (\i. x [^] i) (rev [0..< Suc n])"

  assume x: "x \ F" then have in_carrier: "x \ carrier R"
sing subringE assms2 by
  obtain n where n: " assumes "subfield "FR and" K F"
    using) auto
  have set_Us: "set (proof-
    sing3,)OF byinduct
  hence "set (?Us n) \ carrier R"
    usingassume x: "x \  have in_carrier "x \
  moreover have "dependent K (?Us n)"
    using independent_length_le_dimension[OF assms(1) n _ set_Us] by auto
  ultimately
obtain where:length  " ?n \" ⊆ { 0 }"
    using dependent_imp_non_trivial_combine[OFhave set_Usset n \subseteq"
  have "set Ks \ carrier R"
    using subring_props(1)[OF assms(1)] Ks(3) by auto 
  hence " (normalize Ks) x = \java.lang.StringIndexOutOfBoundsException: Index 41 out of bounds for length 41
    using combine_eq_eval[of Ks] eval_normalize[OF _ in_carrier] Ks(1- subringE)[OF(2)]  auto
moreover"normalize Ks = [] \ setKs \ {\zero> }
    by (using[OF assms)n_set_Us
                    metis
  hence" Ks \ []"
    usingKs4)  ( list.sizenat(1) set_empty)
  moreover have "normalize Ks \ carrier (K[X])"
    using normalize_gives_polynomial subring_propsOF assmsKs) by auto
  ultimatelyshow
    using by auto
qed

corollary (in domain) simple_extension_dim:
  assumes K R " carrier R""algebraic over K) x"
  shows "(dim over K) (simple_extension K metis all_not_in_conv list.discI list.sel(3)singletonD )
  using dimI "normalizeKs \noteq ["

corollary (in domain) finite_dimension_simple_extension:
assumes KR"x\in carrier Rjava.lang.StringIndexOutOfBoundsException: Index 44 out of bounds for length 44
  shows "finite_dimension K (simple_extension K x) show ?thesis
  using finite_dimensionI[OF dimension_simple_extension[OF assms]]
        finite_dimension_imp_algebraic[OF _ simple_extension_is_subring[OF subfieldE(1)
        simple_extension_mem[OF subfieldE(1)] corollary (ndomain:
  by auto


subsectionFinite

lemma (in ring) finite_extension_consistent:
  assumesK R"shows ".finite_extension>carrier\rparr=finite_extension
proof -
  have "\K' xs. ring.finite_extension (R \ carrier := K \corollary (n domain) finite_dimension_simple_extension:
  proof
    K' "ring.( carrier: K \<> 'xsxs
      using    finite_dimensionI dimension_simple_extension assms
            simple_extension_consistent assms(induct xs)
  qed
  thus?hesis blast
qed

lemma (in ring
  assumes "K \ K'" shows ‹
  using mono_simple_extension assms byjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

lemma (in ring) finite_extension_carrier:
  assumes "set xs \🚫
  using  by ( xsauto

lemma (in ringthus ? by blast
  assumes "K \ carrier R" and
  using simple_extension_in_carrierinduct xs)()

emma finite_extension_subring_incl
  assumes "subring mono_simple_extension assmsby( xs) (auto)
  using ring.finite_extension_in_carrier[OF subring_is_ring[OF (in ring) finite_extension_carrier
   finite_extension_consistent simp

lemmain) finite_extension_incl_aux
  assumes "K \ carrier R" and (in ring:
  shows Kxs finite_extensionK (#xs
  using[OF finite_extension_in_carrier assms,) assmsbyjava.lang.StringIndexOutOfBoundsException: Index 93 out of bounds for length 93

lemma (in ring) finite_extension_incl:
  assumesK\subseteq R and "setxs\subseteq>carrierR shows K<> finite_extension Kxsjava.lang.StringIndexOutOfBoundsException: Index 114 out of bounds for length 114
  using[OFassms]assmsby( xs) (auto

lemma (in ring)  ( ring:
\>  "\>carrier" xs carrier R"
  shows "finite_extension K (x # xs) = (\finite_extension 🚫
using[OF[OFa(1,3)] assms simp

lemma (in domain) finite_extension_is_subring:
  assumes "subring K R" "set xs \ carrier R" shows "K \ and setsubseteq shows<   xs
  using assmsusing[OF assms(1) assmsby(nduct xs auto

corollaryin) finite_extension_mem
  assumes subring
  shows "set xs \< "finite_extensionx #) = (<lambdaeval p ((finite_extension xs) [])"
proof induct)
  case Nil
  then show (in)finite_extension_is_subring
next
  case (Cons a xs)
  from Cons(2) have a: "a \ carrier R" and xs: "set xs \ carrier R" by auto
showcase
  proof
    fix x assume "x \ set (a # xs)"
    then consider "x = a" | "x \ set xs" by auto
    then show "x \ finite_extension K (a shows "setsubseteq <>set<ubseteq"
    proof
      case 1
      with a have "x \ carrier R" by simp
      with xs havenext
        using[OFfinite_extension_is_subring[OF]] bysimp
      with 1 show Cons have a:"a \in R"and: "setxs \ carrier R"byauto
    next
      case 2
      with Cons have *: "x \ finite_extension K xs" by auto
      from a xs have "finite_extension K xs \ finite_extension K (a # xs)"
        by (rule finite_extension_incl_aux[OF subringE(1)[OF subring]])
      with * show ?thesis
    qed
  qed
qed

corollary (in domain) finite_extension_minimal:
  assumes "subring K R" "set xs \ carrier R"
  showsfinite_extension <>{K' subring K R \and>K\ \andsetsubseteqK'}java.lang.StringIndexOutOfBoundsException: Index 116 out of bounds for length 116
  usingnext
        finite_extension_incl[OF subringE(1)[OF assms(1)] assms(2)] finite_extension_subring_inclcase java.lang.StringIndexOutOfBoundsException: Index 12 out of bounds for length 12
  by blast

corollary (in domain) finite_extension_same_set:
  assumes "subring K R" "set xs \ carrier R" "set xs = set ys"
  shows "finite_extension K xs = finite_extension K ys"
   finite_extension_minimal(1)]assms  auto

text
proposition)finite_extension_is_subfield
  assumes "subfield K R" "set xs \ carrier R"
  shows "(\x. x \ set xs \ (algebraic overshows=\ { K'.subringK \ K K < xs \subseteq "
  using simple_extension_is_subfield algebraic_mono assms
  by( xs (uto, metis.simps subring_props)

proposition (in domain) finite_extension_finite_dimension:
  assumes "ubfield K R"set ⊆
  shows "(\x. x \ set xs \ (algebraic over K) x) \
    andfinite_dimensionfinite_extension < \And.x\in  xs (algebraic x"
proof -
  show "finite_dimension K (finite_extension K xs) \ (\x. x \ set xs \ (algebraic over K) x)"
    using finite_dimension_imp_algebraic[OF assms(1)
          finite_extension_is_subring[OF subfieldE(1)[OF assms(1)] assms(2)]]
          finite_extension_mem[OF subfieldE(1)[OF assms(1)] assms(2)] by auto "(\And>. x setxs \Longrightarrow>( over K) x) \ subfield (finite_extension K xs) R"
next
  show "(\Andx. x xs \ (algebraic over K) x) \ finite_dimension K (finite_extension K xs)
    by( xs) (auto finite_extensionsimps subring_props
  proof (induct (in) finite_extension_finite_dimension:
    case( x xs
    hence "finite_dimension K (finite_extension K xs)"     "finite_dimension K ( K xs) \\<>x. x \<> set xs \<> (algebraic over K)x"
      by auto
    moreover"algebraicover (inite_extension xs) x"
      using     using finite_dimension finite_dimension_imp_algebraic assms
    moreover "subfield finite_extensionKxs Rjava.lang.StringIndexOutOfBoundsException: Index 54 out of bounds for length 54
      using finite_extension_is_subfield[ assms()]C(2-3) by auto
    ultimately show ?case
      using telescopic_base_dim(1)[OF assms
dimensionI, _x] 3 byauto
  qed
qed

corollary (in (induct, simp: finite_dimensionI dimension_one assms])
  assumes "subfield K R" "set xs \ carrier R" and "\x. x \ set xs \ (algebraic over K) x"
   "y \finite_extension K xs \ (algebraic over K) y"
  usingby auto
        finite_extension_is_subring[OF subfieldE(1)[OF assms(1)] assms(2    moreover have( over) x"
        finite_extension_finite_dimension(1[OF assms) assmsby 

corollary (in domain) simple_extesion_mem_imp_algebraic[OF(1)]Cons3  auto
  assumes "subfield K R" "x \ carrier R" "(algebraic over K) x"
  shows "y \ simple_extension K x \ ifinite_dimensionI[OF dimension_simple_extension, of_x] Cons()by auto
  using finite_extesion_mem_imp_algebraic[OF assms(1), of "[ x qed


subsection ‹

text ‹We show that theassumessubfield " xs \ carrier R"andAndx. x ∈ set xs ==> over)x
      over (1)[OF(1-2)]assms) by

lemma (in field) subfield_of_algebraics:
  assumes K R" shows "subfieldincarrier R. (algebraic over K) x } R"
proof -
  let ?et_of_algebraics x\carrier K)x}"

  show ?thesis
  proof
    show "?set_of_algebraics \ carrier R" and "\ \ ?set_of_algebraics"
      using algebraic_self[OF _ subringE(3)] subfieldE(1)[OF assms ‹ numbers a field
  next
    fix x y assume x: "x \ over K is a subfield itself\<>
    have "\ x \ simple_extension K x"
      using subringE(5)[OF simple_extension_is_subring[OF "subfield K R"shows{x 🚫
            simple_extension_mem1] assmsx by auto
    thus "\ x \ ?set_of_algebraics"
      using simple_extesion_mem_imp_algebraic[OF assms (rule'[OF subringI])

    have "x \ y \ finite_extension K [ x, y ]" and "x \ y \ finite_extension K [ x, y ]"
      using (6-7)[OF finite_extension_is_subring subfieldEOF(1)]] of ,y]]
            finite_extension_mem
     x\oplus y∈ y ∈ ?set_of_algebraics
      using finite_extesion_mem_imp_algebraic[OF assms, of "[ x, y ]"] x y by auto "\x\in simple_extension K x"
 next
    fix z simple_extension_mem(1) assms( java.lang.StringIndexOutOfBoundsException: Index 68 out of bounds for length 68
    have "inv z \ simple_extension K z"
      using subfield_m_inv(1)[    havex \oplusy\in finite_extension,y "and" <>y \in finite_extension,y "
            simple_extension_is_subfield assms z]
            simple_extension_mem[OF subfieldE(1)] assms(1) z by auto
    thus            finite_extension_mem[OF (1)[OF assms(1)], of[ x y ]x y byauto
      using[OF assms z by auto
  qed
qed

end