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products/sources/formale Sprachen/Isabelle/HOL/ex/ (Beweissystem Isabelle Version 2025-1^©) Datei vom 16.11.2025 mit Größe 5 kB

Quelle Finite_Extensions.thy Sprache: Isabelle

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

theory    Author java.lang.StringIndexOutOfBoundsException: Index 24 out of bounds for length 24
   Embedded_Algebras Polynomial_Divisibility in) transcendental 'set

begin

section \<open>Finite Extensions\<close>

subsection \<open>Definitions\<close>

definition (in ringjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
   " K x =(THE p. p\carrier([] \ pirreducible K p \ eval p x = \ \ lead_coeff p = \)"

abbreviation (in ring)java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
  where "algebraic K x \ \ transcendental K x"

definition( ringIrr"aset\java.lang.StringIndexOutOfBoundsException: Index 75 out of bounds for length 75
   "Irr Kx THEp java.lang.StringIndexOutOfBoundsException: Range [0, 66) out of bounds for length 58

inductive_set (in   "K
  java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
    zero  java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
    assumes "transcendental = .transcendental R .[OFsubring_is_ring[assms]]

(ring
  where "

subsection "java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

lemma (in  transcendental_consistentassmslemma )java.lang.StringIndexOutOfBoundsException: Index 36 out of bounds for length 36
" showstranscendental=ringR
unfolding.[OF assms
            univ_poly_consistent[OFhave
   "]
lemma ( ring:
  assumes "subring K R" shows "algebraic ?thesis
   over_def auto:

lemma (in ring    usingassms transcendental_def
java.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3
-
  unfolding K) lambda>p. eval p x) \<noteq> { [] } \<Longrightarrow> (algebraic over K) x" unfolding
byauto : univ_poly_defassumesusing over_def
  thus
   transcendental_def : univ_poly_zero
qed

  ass   "
"java.lang.StringIndexOutOfBoundsException: Range [25, 24) out of bounds for length 103
  using java.lang.StringIndexOutOfBoundsException: Index 13 out of bounds for length 0

lemma (in ring) non_trivial_ker_imp_algebraic [OF]  over_def
  shows
  unfolding by

(domainjava.lang.StringIndexOutOfBoundsException: Index 49 out of bounds for length 49
-
  shows "a_kernel (K[X]) R (\p. eval p x) = { [] } \ (transcendental over K) x"
   ring_hom_ringOF eval_ring_hom[OF assms]]  have "[ in> a_kernel (K[X]) R (\p. eval p x)"
  unfolding transcendental_def over_def by (     a_kernel_def

lemma (in domain)using[ ] unfoldingblast
  assumes "subring K
shows )
using[OFpjava.lang.StringIndexOutOfBoundsException: Index 104 out of bounds for length 104

lemma (using[OFassms' by over K)"
  "and "\in  R"algebraic ) java.lang.StringIndexOutOfBoundsException: Index 70 out of bounds for length 70
  "
proof -
  have "[ java.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3
    unfoldinguniv_poly_def
  then obtain p java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
using
  lemma( ring:
qed

lemma (in   "K \ K'" "(transcendental over K') x" shows "(transcendental over K) x"
mesalgebraicI ]   by java.lang.StringIndexOutOfBoundsException: Index 72 out of bounds for length 72
using  unfolding' auto

lemma (in    using  over_def (etisalgebraicI
  assumes
proof  () java.lang.StringIndexOutOfBoundsException: Index 35 out of bounds for length 35
havecarrier
    using  thus one     auto)
  thus
     assms transcendental_def     using assms unfolding over_def transcendental_def by (metis inj_on_subset indomain)zero_is_algebraic
qed

( ring:
  assumes "
usingOF

lemma (in " K R" "k K" shows "(algebraic over K) k"
     subringE3[ assmsunfolding  by auto  thus
\<open>Minimal Polynomial\<close>

(in subfieldxjava.lang.NullPointerException
  assumes ">!p. minimal_poly p)
algebraicI
  show "[ univ_poly_is_principalOF subringE)OFassms(1)]assms)auto
    using      (, )
  have (ndomain:
    using subringE" K Rjava.lang.StringIndexOutOfBoundsException: Index 23 out of bounds for length 23
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
auto
qed

lemma (
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
" K]R(p eval
proof -
  have "eval [ \ ] x \ \" and "[ \ ] \ carrier (K[X])"
    subringEOF]unfolding  polynomial_defbyuto
  thus ?thesis
    unfolding  hence?minimal_poly
qed

subsection \<open>Minimal Polynomial\<close>

lemma (in domain) minimal_polynomial_is_unique:
  assumes  -
moreover java.lang.StringIndexOutOfBoundsException: Index 67 out of bounds for length 66
    (hencejava.lang.NullPointerException
proof
interpret:principal_domainjava.lang.StringIndexOutOfBoundsException: Range [39, 40) out of bounds for length 39
    using univ_poly_is_principalusing q by

  usingOF OF(2]
java.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3

              ](1)OF1]by
and    >andIrr java.lang.StringIndexOutOfBoundsException: Index 72 out of bounds for length 72
          algebraic_imp_non_trivial_ker[OF_ assms(2-3)]
ker_diff_carriersubfieldE[Fassmsby java.lang.StringIndexOutOfBoundsException: Index 61 out of bounds for length 61
  hence "?minimal_poly p"
    using -
  moreover havefixassume:"? java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
      sing a_kernel_def
    fix q assume q: "?minimal_poly q"
    then have "q deal_pirreducible assumes"subfield"and"<in> carrier R" "(algebraic over K) x"
      using p unfolding a_kernel_def
"p \\<^bsub>K[X]\<^esub> q"
a kera_kernel\eval
    using[OF(1      UP qp simp
       UP q p         unique
    thus "q = p"
      usingker_diff_carrier(
qed Irr
  ultimately show IrrE
qed

lemma (in domain) IrrE:
a subfieldand
  shows "Irr K x \ carrier (K[X])" and "pirreducible K (Irr K x)"
    andlead_coeff  x
          associated_iff_same_ideal[OF(1)[OF]java.lang.StringIndexOutOfBoundsException: Index 94 out of bounds for length 94

inIrr_generates_ker

  shows
( ) :
  obtain q
     a_kernel<
      and ker "p
     exists_unique_pirreducible_gen()[OF()
          [OF_      q q
                kerKX]
havex <in> PIdl\<^bsub>K[X]\<^esub> q"
usingIrrE,)OF] ker a_kernel_def
  thus ?thesis
     cgenideal_pirreducible(1 q1)(2)[Fassms (1)[ assms
          cring" x \^>[\java.lang.StringIndexOutOfBoundsException: Index 49 out of bounds for length 49
    unfolding ker thesis
    by simp
qed

lemma (in domain) Irr_minimal:
subfield x \<in> carrier R" "(algebraic over K) x"
    and "p \ carrier (K[X])" "eval p x = \" shows "(Irr K x) pdivides p"
-
interpret: "K[]java.lang.StringIndexOutOfBoundsException: Index 39 out of bounds for length 39
    univ_poly_is_principal(1)].

  have
lemma in) simple_extension_consistent
  hence "(Irr "subringR  ". R \ carrier := K \) = simple_extension"
     " K K'" shows "simple_extension K x \ simple_extension K' x"
( UP UP assms)
  thus ?thesis
    unfolding pdivides_iff_shell[OF assmscring[OFuniv_poly_is_cring[OF subfieldE[ proof
qedbjava.lang.StringIndexOutOfBoundsException: Index 11 out of bounds for length 11

lemma (in domainqed
   "subfieldKR xjava.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3
using[OF assms    " carrier(KX) evalpx=\" shows "(Irr K x) pdivides p"

subsection \<open>Simple Extensions\<close>

lemma (in ring) simple_extension_consistent:
  assumesinterpretjava.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 5
proof -
  interpret [OF(2)] subringE)OFpdivides_iff_shell ()IrrE assms13 assms]java.lang.StringIndexOutOfBoundsException: Index 79 out of bounds for length 79
    using subring_is_ring[OF assms]   using   hus

  have
s \<open>Simple Extensions\<close>
     (ring:
      by (induction R" subring R ". Rjava.lang.StringIndexOutOfBoundsException: Index 107 out of bounds for length 107
  qed
  moreover
K' x<>K. K'xjava.lang.StringIndexOutOfBoundsException: Index 79 out of bounds for length 79
  proof
   "simple_extension (carrier R) x\ carrier R"
   \And'x K'x\<subseteq> K.simple_extension  K' x"
       induction.) (simp ring[OF subring_is_ring assms) assms3
  qed
  ultimately ?thesis
qed

lemma(
  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 (induct a rule: simple_extension.induct, simp)
    casejava.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 5
  qed
qed

java.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3
  assumesjava.lang.StringIndexOutOfBoundsException: Index 102 out of bounds for length 102
proof
   kassumek < "thus" \<in> simple_extension K x" -
    simple_extension_incl () 1,)[ ()]byauto
qed

lemma (in ring) simple_extension_mem:
  assumes "subring moreover "set(  [k2 <subseteq> K"
proof -
  have "\ \ simple_extension K x"
    using        usingpolynomial_inclOF p()]\<pen \<in> K\<close> by auto
  thus ?thesis
  thus normalize_gives_polynomial  blast

injava.lang.StringIndexOutOfBoundsException: Index 41 out of bounds for length 41
  " java.lang.StringIndexOutOfBoundsException: Index 80 out of bounds for length 80
proof
show  <subseteq> simple_extension (carrier R) x"simple_extensionx
    using
next" (arrierR)\java.lang.StringIndexOutOfBoundsException: Index 61 out of bounds for length 61
      fix assume " \ (\p. eval p x) ` carrier (K[X])"
  proof
    fix a        polynomial_incl univ_poly_def
byinduct: simple_extension) (auto add)
  qed
qed

lemma (in ring) simple_extension_in_carrier      case  thuscase
  assumes "K \ carrier R" and "x \ carrier R" shows "simple_extension K x \ carrier R"
  using mono_simple_extension[OF assms        using simple_extension by simp

lemma (inobtain' wherep "p =p @[k "
   "subring K' R"andK\<subseteq> K'" "x \<in> K'" shows "simple_extension K x \<subseteq> K'"
  using ring.simple_extension_in_carrier[OF          eval_append_auxof'kx]Suc3- unfolding p by auto
  unfolding simple_extension_consistent[OF assms(1)] by simp

lemma (in ring) simple_extension_as_eval_img:
  assumes "K \ carrier R" "x \ carrier R"
  shows "simple_extension K x = (\p. eval p x) ` carrier (K[X])"
proof
  showsimple_extensionaassumea \<in> simple_extension (carrier R) x" thus "a \<in> carrier R"
  proof
fixaassume" simple_extension.lin Suc3)unfolding pbyauto
( .)
      java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
have  ["and" [x=\<>
        unfolding  by simp
     thus
          using ring
    next (in) simple_extension_minimal
  assumes  "x \ carrier R"
      then obtain p whereshowssimple_extensionx=
         auto add)
      hence "set p \ carrier R" and "k2 \ carrier R"
using(1) (2) unfoldingpolynomial_def
      hence "eval (normalize (p @ [ k2 ])) blast
        using corollary (in domain:
      moreover have "set (p <> carrier R"
        using[OF p2) <open>k2 \<in> K\<close> by auto
      then have " ring_hom_ring.FactRing_iso_set_aux[F [OF assms]]
        using normalize_gives_polynomial univ_poly_carrier by blast
      ultimately [OF(1[ assms(2)]
        unfolding by force
    qed
  qed
next
    assumessubfieldR andx\<in> carrier R" "(algebraic over K) x"
  proof
    fix (1)[  unfolding simple_extension_consistent assms bysimp
then pwhere" p \ K" "eval p x = a"
      using polynomial_incl unfolding univ_poly_def by auto
    thuscorollary( domain) simple_extension_of_transcendental
proofinduct arbitrary p a
      case 0 thus
      singzero java.lang.StringIndexOutOfBoundsException: Index 43 out of bounds for length 43
    next
      case(Suc n
      obtainp   : p=p' @ k "
u Suc        ring_iso_trans[OF ringjava.lang.StringIndexOutOfBoundsException: Index 75 out of bounds for length 75
       (
              casezero
      moreover have "eval p' x \ simple_extension K x"
        using Suc(
      ultimately) simple_extension_subfield_imp_algebraic
         simple_extension Suc) p byauto
    qed
  qed
qed

corollary (in domain) simple_extension_is_subring  shows" (simple_extension K x) R (algebraic over K) x"
  assumes "subring K R" "x \ carrier R" shows "subring (simple_extension K x) R"
  using ring_hom_ring univ_poly_carrier
        ring.carrier_is_subringjava.lang.StringIndexOutOfBoundsException: Index 8 out of bounds for length 8
        simple_extension_as_eval_img[OF
  bysimp

(indomain simple_extension_minimal
  assumes     obtain  where " \ ring_iso (R \ carrier := simple_extension K x \) (K[X])"
  shows   =java.lang.StringIndexOutOfBoundsException: Index 104 out of bounds for length 104
  using simple_extension_is_subring[     interpret : ring_hom_ringR\lparr  :   x \<rparr>" "K[X]" h
        simple_extension_incl subringE) assms() simple_extension_subring_incl
  byblast

corollary in) :
  assumes "subring K R" "x \ carrier R"
"K[X)Quot(kernel([X \lambdap evalpx) simeq> R \ carrier := simple_extension K x \"
  using ring_hom_ring.FactRing_iso_set_aux[OF eval_ring_hom[OF        " (normalize (p [ ])) x=k1\java.lang.StringIndexOutOfBoundsException: Index 74 out of bounds for length 74
        simple_extension_as_eval_img[OF        fieldring_iso_imp_img_field subfield_iff[ simple_ext]
  unfolding by auto

corollary (in domain) simple_extension_of_algebraic:
assumes R  "x \ carrier R" "(algebraic over K) x"
  shows "Rupt K (Irr K x) \ R \ carrier := simple_extension K x \"
  using simple_extension_isomorphism[OF subfieldE(1)[OF assms(1)  qed
  Irr_generates_ker  simp

corollary (in domain) simple_extension_of_transcendental polynomial_inclOFp()] \<open>k2 \<in> K\<close> by auto
  assumessubring "and" \<in> carrier R" "(transcendental over K) x"
shows   "subfieldK "" < carrier R"
  usingshows"( ) \ (algebraic over K) x"
        ring_iso_trans   obtain  where:"h\<> ring_iso ( K (Irr K ) (
  unfoldingusingsimple_extension_of_algebraic assms]unfolding by blast
  byjava.lang.StringIndexOutOfBoundsException: Index 9 out of bounds for length 9

proposition   interpret: ring_hom_ringK( Kx"" \<lparr> carrier := simple_extension K x \<rparr>" h
assumesK""x\ carrier R"
  shows "subfield (simple_extension K x) R \ (algebraic over K) x"
proof-
  assume simple_ext: "subfield (simple_extension K show "subfieldsimple_extension "
  proof     field.ring_iso_imp_img_field[OFrupt_field subfield_iff[ _
x" then have "(transcendental overxjava.lang.StringIndexOutOfBoundsException: Index 78 out of bounds for length 78
      unfolding over_def by simpassumesimple_extsubfieldsimple_extensionx "thus"algebraic K) x"
     obtainh  h: "h ing_iso ( \ carrier := simple_extension K x \) (K[X])"
      java.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3
      unfolding is_ring_iso_def (indomain exp_base_independent
theninterpretHomring_hom_ring <lparr> carrier := simple_extension K x \<rparr>" "K[X]" h
      using subring_is_ring[OF simple_extension_is_subring[OF assms]]
            univ_poly_is_ring[OF assms(1)] assms h
ring_hom_ring_axioms_def
    haveproof
using.[OF() simple_extjava.lang.StringIndexOutOfBoundsException: Index 77 out of bounds for length 77
      unfolding Hom.hom_one Hom.hom_zero by simp
moreover \not  KX)
      using univ_poly_not_field[OF assms(1)] .
    ultimately show False by simp
  qed
qed

proposition (in domain) simple_extension_is_subfield "<> [^]n\notin> K (exp_base x n)java.lang.StringIndexOutOfBoundsException: Index 62 out of bounds for length 62
  assumes "subfield K R" "x \ carrier R"
  shows            Span_mem_imp_non_trivial_combine assms [OFassms of n]
proof
  assume alg:        hence" (a # Ks) =\java.lang.StringIndexOutOfBoundsException: Index 41 out of bounds for length 41
t obtainh  h: " \ ring_iso (Rupt K (Irr K x)) (R \ carrier := simple_extension K x \)"
               univ_poly_def using Ks-2) by
have: " (Rupt K (Irr Kx)"and ( \<lparr> carrier := simple_extension K x \<rparr>)"
    using subring_is_ring[OF           pdivides_imp_degree_le[OF(1)[OF (1)]
rupture_of_Irr assmsalg by simp
  then interpret Hom: ring_hom_ring "Rupt K (Irr K x)" " \Suc n \ degree (Irr K x)\ and this show False by simp
    using h cring.axioms(1)[OF domain.axioms         independent.
         simp
  show "subfield (simple_extension K x) Rqed
java.lang.StringIndexOutOfBoundsException: Index 2 out of bounds for length 0
[OF(3)[OF(1)] assms
    by 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)[    is? =?"java.lang.StringIndexOutOfBoundsException: Index 28 out of bounds for length 28
qed

subsection      obtain whereKs" Ks \ K" "length Ks = n" "u = combine Ks (exp_base x n)"

lemma (in domain) exp_base_independent:
  assumes "subfield K R" "x \ carrier R" "(algebraic over K) x"
  shows "independent K (exp_base x (degree (Irr K java.lang.StringIndexOutOfBoundsException: Range [0, 51) out of bounds for length 38
proof-
  have " moreover have normalize Ks carrier (K[X])"
  proof
    fix     have "length normalize ) \ n"
    proof ( n, simp: exp_base_def
      case     show " ?eval_img" by auto
      have " qed
      proof (rule ccontr)
        assume "\ x [^] n \ Span K (exp_base x n)"
then Ks
where:" K - { \ }" "set Ks \ K" "length Ks = n" "combine (a # Ks) (exp_base x (Suc n)) = \"in   <zero> }" "set Ks \<subseteq> K" "length Ks = n" "combine (a # Ks) (exp_base x (Suc n)) = \<zero>"
          using  "(\p. eval p x) ` carrier (K[X]) \ simple_extension K x"
          by (auto simp add: exp_base_def "combine p (xp_base x lengthp)) = ujava.lang.StringIndexOutOfBoundsException: Index 49 out of bounds for length 49
        hence "eval (a # Ks) x = \"
          using by (auto add:exp_base_def
    henceset
          unfolding univ_poly_def subfieldE)OF(1)] by auto
have "egree Irr x) \ n"
          using pdivides_imp_degree_le[OF subfieldE(1)[OF       p(2)drop_exp_base
IrrE)OF] _ _  Irr_minimal assms ofa#Ks]Ks by auto
        from \<open>Suc n \<le> degree (Irr K x)\<close> and this show False by simp
      qed
      thus ?case
        using independent.li_Cons assms(2) Suc by (auto simp add: exp_base_def      using combine_prepend_replicate[F  exp_base_closed assms(2, of]
    qed
  qed
  thusthesis
    by simp show " ?Span"
qed

lemma (in ring) Span_eq_eval_img:
java.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3
  shows "Span K (exp_base x n) = (\p. eval p x) ` { p \ carrier (K[X]). length p \ n }"
is  eval_img
proof
  show "?Span \ ?eval_img"
  proof
    fix  unfolding simple_extension_as_eval_imgOF (3[OF assms)]assms
    thenobtainKs Ks: set <subseteq> K" "length Ks = n" "u = combine Ks (exp_base x n)"
      using Span_eq_combine_set_length_version[OF assms(1) exp_base_closed[OF assms(2)]]
      by (auto simp add: exp_base_defproof ()
    hence" eval normalize)xjava.lang.StringIndexOutOfBoundsException: Index 37 out of bounds for length 37
using  eval_normalizeOF (2)]subfieldE(3)[Fassms(1) byauto
    java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
      usinghave: "IrrKx\
moreover "length( Ks) \ n"
      using normalize_length_le[of Ksthen obtain r
    ultimately show "u \ ?eval_img" by auto
  qed
next
  show      q: "q \ carrier (K[X])" and r: "r \ carrier (K[X])"
  proof
    fix u assume "u \ ?eval_img"
    then obtain where " \ carrier (K[X])" "length p \ n" "u = eval p x"
         "eval x a " \<in> (\<lambda>p. eval p x) ` carrier (K[X])"
     " (exp_base (length p)= ujava.lang.StringIndexOutOfBoundsException: Index 49 out of bounds for length 49
thus
    moreover have    proof( "length p :pajava.lang.StringIndexOutOfBoundsException: Index 44 out of bounds for length 44
      using polynomial_incl[of K p] p(1) unfolding univ_poly_carrier by auto
    henceset\<subseteq> carrier R"
      using subfieldE(3     dvd() Irr byauto
    moreover have "drop (n - length p) (exp_base x n) = exp_base x (length p)"
using(drop_exp_base
    ultimatelyshow  where[k]java.lang.StringIndexOutOfBoundsException: Index 43 out of bounds for length 43
using[  [OF
    moreover
       subringE2[F (1)[OF (1)]] set_p
ultimately java.lang.NullPointerException
              (1-)  p by
  qed dimension (Kx)K(Kx"
qed

lemma (in domain) Span_exp_base:
   "subfield KR"" \ carrier R" "(algebraic over K) x"
K x)) =java.lang.StringIndexOutOfBoundsException: Index 61 out of bounds for length 61
  unfoldingcorollary (domainsimple_extension_is_subring
            Span_eq_eval_img
proof (autousing .img_is_subring eval_ring_hom assms
  interpret:  "K[X]java.lang.StringIndexOutOfBoundsException: Index 39 out of bounds for length 39
    using univ_poly_is_principal[OF assms(1)] .
  note hom_simps = ring_hom_memE (in ring finite_dimension_imp_algebraic

  fix p assume p: "p \ carrier (K[X])"
  have Irr: "Irr K x \ carrier (K[X])" "Irr K x \ []"
using IrrE-[F ] unfolding ring_irreducible_defuniv_poly_zero by auto
  then obtain q r
    where q " \ carrier (K[X])" and r: "r \ carrier (K[X])"
      and java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
     subfield_long_division_theorem_shell assms) pIrr] unfolding by auto
  obtain where n:"dimension n K F"
    using hom_simps assms byauto
  hence "eval p x =eval r x"
    using     x subringE6[ assms() by inductauto
  hence "et(Us n)\
dvdIrrauto
  ultimatelyassume "\ (algebraic over K) x" then have "(transcendental over K) x"
u over_defbysimp
    using r by auto
qed

corollaryin)dimension_simple_extension
  assumesfinite_dimension_imp_algebraic [OFsubfieldE
  shows "dimension (degree (Irr K is_ring_iso_def by java.lang.StringIndexOutOfBoundsException: Index 40 out of bounds for length 40
  using dimension_independent
   ( add subsection

ring
  assumes "subfield using ring.finite_extension_in_carrier[OF "independentexp_base (IrrK))
  showshave\Andn.n\<le> degree (Irr K x) \<Longrightarrow> independent K (exp_base x n)"

   in ( ccontr ccontr)

   x " \ F" then have in_carrier: "x \ carrier R"
    using subringE[OF assms(           Kss " Kxs\subseteq>inite_extensionK ( #xs)
  obtain where :" n KF"
    using assms(3) by auto
:set
using(3,6)[ assms]byinduct)
   " (? have "degree )lejava.lang.StringIndexOutOfBoundsException: Index 50 out of bounds for length 50
()[OF]byauto
  moreoverhavedependent? "
    using independent_length_le_dimension[OF assms(1) [1assms ,  "#] ()auto
  ultimately
  :
    using dependent_imp_non_trivial_combine : "subringKR"
  "set \ carrier R"
    using subring_props(
java.lang.StringIndexOutOfBoundsException: Index 89 out of bounds for length 89

  moreover have     x assume " set ( #xs)"
by(Ks, .,
                    metis all_not_in_conv list.discI list.selusing[OF    then "x =a" x<in> set xs" by auto
normalizehave"
    using Ks      axsKxs
       normalize_length_le KsKsbyauto
    using normalize_gives_polynomialOF Ks(]unfoldinguniv_poly_def auto
  ultimately ?thesis
    java.lang.StringIndexOutOfBoundsException: Range [0, 9) out of bounds for length 5
qed

corollary java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
   idomain:
  "java.lang.StringIndexOutOfBoundsException: Index 55 out of bounds for length 55
    moreover <open>The reciprocal is also true, but it is more subtle.\<close>

corollary (in   havecombinen- p
  shows
  showsfinite_dimensionsimple_extension)java.lang.StringIndexOutOfBoundsException: Index 135 out of bounds for length 135
java.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3
finite_dimension_imp_algebraic _simple_extension_is_subring (1)]]
        " (<>. x\ setxs\

\<open>Finite Extensions\<close>

lemma (in ring)  note = ring_hom_memE[   : "IrrKx\carrier (KX) ualgebraic_mono[ finite_extension_inclFsubfieldE)[ (1]] 23 java.lang.StringIndexOutOfBoundsException: Index 100 out of bounds for length 100
  assumes   obtain q r
proof -
  have " ultimately show ?case
  proof-
    fixK'xs show "ringfinite_extension ( \ carrier := K \) K' xs = finite_extension K' xs"
       ring.simpsOF[F ]]]
            simple_extension_consistent[  qed
  qed
  thus      xsubringE6 assms OFxby
qed

  ):
simple_extension_is_subfield assmsz]
  using mono_simple_extension assms by (induct xs) (auto)

lemma(n ring) finite_extension_carrier
  assumes"et metisall_not_in_convlistdiscI sel(3) singletonDsubset_singletonD))
usingsimple_extension_carrierxsauto

( ring:
  assumes "K \ carrier R" and "set xs \ carrier R" shows "finite_extension K xs \ carrier R"
  using assms simple_extension_in_carrier by (induct havenormalize

lemma (in ring) finite_extension_subring_incl:
  assumes "subring K' R" and " algebraicI by java.lang.StringIndexOutOfBoundsException: Index 28 out of bounds for length 28
  using ring.finite_extension_in_carrier[OF subring_is_ring[OF assms(1)  assumes "subfieldK ""x <> carrier " "algebraic over K) xjava.lang.StringIndexOutOfBoundsException: Index 67 out of bounds for length 67
u finite_extension_consistent assmssimp

lemma (in ring) finite_extension_incl_aux:
  assumes "Kjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
  shows   assumesK R"x
simple_extension_incl [OF(13]assms) bysimp

lemma (in ring) finite_extension_incl:
  assumes "K \ carrier R" and "set xs \ carrier R" shows "K \ finite_extension K xs"
  using[OF assms)]assmsby induct) (uto)

lemma in) finite_extension_as_eval_img
  assumes "K \ carrier R" and "x \ carrier R" "set xs \ carrier R"
  shows "finite_extension K (x # xs) = (\p. eval p x) ` carrier ((finite_extension K xs) [X])"
  using[OFfinite_extension_in_carrier[OF(1,3)] assms2] byy simp

lemma (in domain) finite_extension_is_subring:
  java.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 0
  using assms lemma (in ring) finite_extension_consistent

corollary (in domain) finite_extension_memproof -
  assumes subring: "subring K R"
  shows "set xs \ carrier R \ set xs \ finite_extension K xs"
proofduct)
  case Nil
  thenshowcasebysimp
next
  case (Cons a xs)
  from Cons(2) havethus by blast
  show ?case
  proof
    fixjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
    thenconsider " a" |x \<in> set xs" by auto
    then   usingmono_simple_extension assms by ( xs) ()
    proof cases
      case 1
      with a have "x \ carrier R" by simp
      with xs have "x \ finite_extension K (x # xs)"
        using[OF [OF subringbysimp
      with 1 show ?thesis by simp
    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   "K \ carrier R" and "set xs \ carrier R" shows "finite_extension K xs \ carrier R"
      with* show ?thesis by auto
    qed
  qed
qed

(in) finite_extension_minimal:
  assumes "subring K R" "set xs \ carrier R"
  showsfinite_extensionxsjava.lang.StringIndexOutOfBoundsException: Index 116 out of bounds for length 116
  using finite_extension_is_subring[OF assms] finite_extension_mem[OF assms]
        finite_extension_incllemma( ring) finite_extension_incl_aux
  by blast

corollary (in domain) finite_extension_same_set:
  assumes "subring K R" "set xs \ carrier R" "set xs = set ys"
  shows "inite_extension xs =finite_extensionK
using[assms(2-) java.lang.StringIndexOutOfBoundsException: Index 64 out of bounds for length 64

text \<open>The reciprocal is also true, but it is more subtle.\<close> in) finite_extension_as_eval_img
proposition (in domain) finite_extension_is_subfield:
  assumes "subfield K R" "set xs \ carrier R"
  shows "\x. x \ set xs \ (algebraic over K) x) \ subfield (finite_extension K xs) R"
  using simple_extension_is_subfield algebraic_mono assms
  by (induct xs) (auto, metis finite_extensionlemma (in

proposition (in domain) finite_extension_finite_dimension:
  assumes "subfield K R" "set xs \ carrier R"
shows
    and "finite_dimension K (finite_extension K xs) \ (\x. x \ set xs \ (algebraic over K) x)"

  show "finite_dimension assumessubring " K R"
    usingfinite_dimension_imp_algebraic assms1
          finite_extension_is_subring subfieldE[ assms1] assms]java.lang.StringIndexOutOfBoundsException: Index 77 out of bounds for length 77
          finite_extension_memthen ?case bysimp
next
  show "(\x. x \ set xs \ (algebraic over K) x) \ finite_dimension K (finite_extension K xs)"
    next
  proof (induct xs, simp add: finite_dimensionI (Cons xs
    case( x xs)
    hence "finite_dimension K (finite_extension K xs)"
      by auto
    moreover have "(algebraic over (finite_extension K xs)) x"
using[OFfinite_extension_inclOFsubfieldE(3)[OF(1)]]] Cons-)by
         show "
      using finite_extension_is_subfield[OF assmsproof cases
    ultimately show ?case
      using telescopic_base_dim(1)[OF assms(1) _ _
       1
  qed
qed

java.lang.StringIndexOutOfBoundsException: Index 56 out of bounds for length 56
  "subfield K R" " xs carrier R" and "\x. x \ set xs \ (algebraic over K) x"
  shows "y \ finite_extension K xs \ (algebraic over K) y"
  using finite_dimension_imp_algebraic[OF assms(1)
        finite_extension_is_subring[OF subfieldE(1)[OF assms(1)] assms(2)]]
ite_dimension(1)[ assms(-) (3) by auto

corollary (in domain) simple_extesion_mem_imp_algebraic:
  assumes "subfield K R" "x \ carrier R" "(algebraic over K) x"
  showsy \<in> simple_extension K x \<Longrightarrow> (algebraic over K) y"
  using finite_extesion_mem_imp_algebraic[OF assms(1), of "[ x ]"] assms(2-3) by auto

subsection

text \<open>We show that the set of algebraic numbers of a field
       a subfield a ubfield.\<close>

lemma (in field) subfield_of_algebraics:
  assumes "subfield K R" shows "subfield { x \ carrier R. (algebraic over K) x } R"
proof -
  let ?set_of_algebraics = "{ x "finite_extensionKxs <>{K' subringK' R\<and> K \<subseteq> K' \<and> set xs \<subseteq> K' }"

  show ?thesis
  proof( subfieldI'OFsubringI])
byblast
      using algebraic_self[OF _ subringE(3)] subfieldE(1)[OF assms(1)] by auto
  next
    fix  yassume x " ?set_of_algebraics" and y: "y \ ?set_of_algebraics"
    have "\ x \ simple_extension K x"
      subringE5[ simple_extension_is_subring subfieldE1]java.lang.StringIndexOutOfBoundsException: Index 72 out of bounds for length 72
            [OF subfieldE(1) assms)x byauto
    thus "\ x \ ?set_of_algebraics"
      using simple_extesion_mem_imp_algebraic[OF assms] x by auto

    have "x \ y \ finite_extension K [ x, y ]" and "x \ y \ finite_extension K [ x, y ]"
      usingpropositionin) finite_extension_is_subfield
finite_extension_memsubfieldEOF1,of , y ]"] x y byauto
    thus "x \ y \ ?set_of_algebraics" and "x \ y \ ?set_of_algebraics"
      using finite_extesion_mem_imp_algebraic[OF assms, of "[ x, y ]"] x y by auto
  next
    fix  by(induct xs (, metis .simps finite_extension_incl(1))
    have "inv z \ simple_extension K z"
using(1)[of"simple_extension K z"]
            simple_extension_is_subfield[OF assms, of z]
            simple_extension_mem[OF subfieldE(1)] assms(1) z by auto
    thus "inv z \ ?set_of_algebraics"
      usingsing simple_extesion_mem_imp_algebraicOF assmsfield_Units auto
  qed
qed

end

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