Impressum Finite_Extensions.thy Sprache: Isabelle

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

theory
  importsimportsEmbedded_Algebras  Polynomial_Divisibility

begin

section java.lang.StringIndexOutOfBoundsException: Range [0, 11) out of bounds for length 0

subsectionimports Polynomials

definition( ringtranscendental ::"a \ 'a \ bool"
java.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 5

abbreviation (in ring) algebraic  java.lang.StringIndexOutOfBoundsException: Index 11 out of bounds for length 0
  where "algebraic K x \ \ transcendental K x"

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

inductive_set (in
  for K andwhere
    zero [simp, intro]: "\ \ simple_extension K x" |
   "finite_extension K xs foldr (\x K'. simple_extension K' x) xs K"

fun ( ring)finite_extension "a set\'a  \<Rightarrow> 'a set"
  wherefinite_extensionK  =foldr

subsection \<open>Basic Properties\<close>

lemma (in ring) transcendental_consistent:
   "subring K R" showstranscendentalring (   transcendental_deftranscendental_def OF

fun(in )  :: "' \ 'a list \ 'a set"

lemma (in ring
  assumesjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
lemma( ring:

lemma (in ring)    "( over K) ""p\in (K[X]) eval px=\zero"shows ]
  assumestranscendental KR"shows"  . ( \<lparr> carrier := K \<rparr>)"   transcendental_def ringtranscendental_def subring_is_ring[OF]]
proof -
have[java.lang.StringIndexOutOfBoundsException: Index 58 out of bounds for length 58
  simp)
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
    using  unfolding over_def inj_on_def by auto
qed

lemma (in ring) transcendental_imp_trivial_ker:
  shows
  usinglemma ( ring:

lemma (in  shows -
   "a_kernel K] (lambda
  using transcendental_imp_trivial_kerjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

     ( simpadd)
   "subring transcendental_imp_trivial_ker over_def auto

.trivial_ker_imp_inj
unfolding bysimp

lemma
umes "subring KR"andx  assumes "subring K R" and "x \<in> carrier R"showstranscendental K  Longrightarrow> a_kernel (K[X]) R (\<lambda>p. eval p x) = { [] }" ring_hom_ring[OF[OF]]
  shows( over  <Longrightarrow> a_kernel (K[X]) R (\<lambda>p. eval p x) \<noteq> { [] }"
  using trivial_ker_imp_transcendental assms unfolding by auto

lemmau transcendental_imp_trivial_ker over_def auto
  lemma(n domain trivial_ker_imp_transcendental
obtainswhere \<in> carrier (K[X])" "p \<noteq> []" "eval p x = \<zero>"
proofjava.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 7
have"]\
unfolding' univ_poly_def byauto
  then obtain p where
    algebraic_imp_non_trivial_kerassms a_kernel_defblast
  thus thesis using that by auto
java.lang.StringIndexOutOfBoundsException: Index 20 out of bounds for length 3

lemma   "(algebraic overK x \ a_kernel (K[X]) R (\p. eval p x) \ { [] }"
  assumes<in> carrier (K[X])" "p \<noteq> []" and "eval p x = \<zero>" shows "(algebraic over K) x"
  using a_kernel_defuniv_poly_def

lemma (  then  p wherejava.lang.StringIndexOutOfBoundsException: Index 83 out of bounds for length 83
ntal)x
proof -
  have "carrier mes"subring x<>carrier"algebraicover K) "
    using assms(1)qed
  thus
     assms pwhere\<in> carrier (K[X])" "p \<noteq> []" "eval p x = \<zero>"
qed

corollary (in ring) algebraic_mono:
  assumes "K \ K'" "(algebraic over K) x" shows "(algebraic over K') x"
   transcendental_monoOF(1)] assms unfoldingunfoldinga_kernel_def byauto

lemma (in domain) zero_is_algebraic:
  assumes "subring K R" shows "(algebraic algebraic_imp_non_trivial_ker[java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
  usingOF1[ assmsvar_defauto

lemma (in domainusing(1) unfolding univ_poly_def by auto
  assumes "subring K Rjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
proof ( [of \<one>, \<ominus> k ]"])
  show "[ \, \ k ] \ carrier (K [X])" and "[ \, \ k ] \ []"
    usingcorollary ring:
  have "k \ carrier R"
    using  have "carrier (K[X]) \ carrier (K'[X])"
"eval [\, \ k ] k = \"
    by(, algebra
qed

lemma (in domain  thus ?thesisusing unfolding over_def by (metisinj_on_subset
"subring
  shows "a_kerneljava.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3
proof -
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
using()OF]  univ_poly_defpolynomial_def
?thesis
    nfolding' by blast
qed

subsection

lemmain domain minimal_polynomial_is_unique:
assumessubfield"and" in> carrier R" "(algebraic over K) x" "algebraicover )"
  shows "\!p \ carrier (K[X]). pirreducible K p \ eval p x = \ \ lead_coeff p = \"
exists?p"
proof -
  interpretproof (rule [of "[\, \ k ]"])
    using[    using1[ assms (   java.lang.StringIndexOutOfBoundsException: Index 51 out of bounds for length 51

  let ?ker_genshows KX)R (\<lambda>p. eval p x) \<noteq> carrier (K[X])"
                    a_kernel (Kproof java.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 7

  obtain p wherethus ?hesis
    using exists_unique_pirreducible_gen     a_kernel_defjava.lang.StringIndexOutOfBoundsException: Index 31 out of bounds for length 22
          algebraic_imp_non_trivial_ker _(2-3   a_kernelX R\lambda.eval
lemma in)minimal_polynomial_is_uniqueusing() assms unfoldinguniv_poly_def byauto
" p"
    using UP.cgenideal_self pjava.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3
have<
  proof -
    fixusing [OF(1) .
    then have "q \ PIdl\<^bsub>K[X]\<^esub> p"
       p  a_kernel_defauto
     "p \<^bsub>K[X]\<^esub> q"
      using -
       UP principal_domain "K[X]"
using.associated_iff_same_ideal by simp
    thus "q = p"
       unique simp

  ultimately[ (1)eval_ring_hom)
qed

( domain) IrrE
  assumes
  shows subfieldE(auto
lead_coeffx=oneevalKx)  <>java.lang.StringIndexOutOfBoundsException: Index 72 out of bounds for length 72
  using           ] (1)[ (1)] byautomoreoverhave "<>.?minimal_poly q <> p

lemma (in haveq\<injava.lang.StringIndexOutOfBoundsException: Index 50 out of bounds for length 50
assumes " carrier R" "(algebraic over K) x"
  shows "a_kernel (K[X]) R (\p. eval p x) = PIdl\<^bsub>K[X]\<^esub> (Irr K x)"

  obtain q
        hence
nd: "a_kernel K[X) R(lambdap.eval )=PIdl\<^bsub>K[X]\<^esub> q"
     exists_unique_pirreducible_gen assms)usingassociated_iff_same_ideal  by
                using.associated_iff_same_idealpby      using q  simp
          ker_diff_carrier] subfieldE  qed
  have" K x \ PIdl\<^bsub>K[X]\<^esub> q"
    using(qed
  thus
     " (Irr Kx)=\" and "eval (Irr K x) x = \"
cring univ_poly_is_cringsubfieldE) (1)]java.lang.StringIndexOutOfBoundsException: Index 94 out of bounds for length 94
    unfolding
    by java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
qed

lemma(ndomainIrr_minimal
assumes" KX)R(p. eval p x) = PIdl\<^bsub>K[X]\<^esub> (Irr K x)"
    and \<in> carrier (K[X])" "eval p x = \<zero>" shows "(Irr K x) pdivides p"
proofalgebraic_imp_non_trivial_ker _assms:"
  interpret ker KX)(\<lambda>p. eval p x) = PIdl\<^bsub>K[X]\<^esub> q"
    using univ_poly_is_principal[OF assms(     (1,[ assms unfolding' by auto

  have "p \ PIdl\<^bsub>K[X]\<^esub> (Irr K x)"
    using Irr_generates_ker[OF assms(1-3)] assmsusing[OF assms)q(-2 IrrE)[F ]] q(1) IrrE[OF]
  hence "Irr K ) divides p"
    using UP.to_contain_is_to_divide IrrE(1)[OF assms(1-3)]
    by (meson UP.cgenideal_ideal UP.cgenideal_minimal assms(4))
  thus?java.lang.StringIndexOutOfBoundsException: Index 14 out of bounds for length 14
    unfolding pdivides_iff_shell[OF   assumes"subfield K R" and"x \ carrier R" "(algebraic over K) x"
qed

proof
   UP principal_domain["
  using rupture_is_field_iff_pirreducible[OF assms(1)] IrrE(1-2)[OF assms] by simp

using[OF assms .

a( ring:
assumes K "showsringsimple_extension(
proof -
  interpret K: ring "R \ carrier := K \"
    using subring_is_ring[OF assms] .

ve\<And>K' x. K.simple_extension K' x \<subseteq> simple_extension K' x"
  proof
    fix K' x a show "a \ K.simple_extension K' x \ a \ simple_extension K' x"
      by (induction rule: K.simple_extension.induct) (autoalgebraic_imp_non_trivial_ker _ assms(2-3)
  qed
  moreover
  have "\K' x. simple_extension K' x \ K.simple_extension K' x"
  proof
    fix K' x a assume a: "a \ simple_extension K' x" thus "a \ K.simple_extension K' x"
      using.simple_extension.zero K.simple_extension
     (induction rule: simple_extension.induct) (simp)+
  qed
  ultimately thesis
qed

lemma (in ring) mono_simple_extension:
  assumes"\java.lang.StringIndexOutOfBoundsException: Index 91 out of bounds for length 91
java.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 5
     java.lang.StringIndexOutOfBoundsException: Index 11 out of bounds for length 11
   ( a rule., simp)
    case lin
  java.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 5

lemma    rupture_is_field_iff_pirreducible(1andcarrier  \<zero>" shows "(Irr K x) pdivides p"
  assumes "K java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
proof
  fix  kjava.lang.StringIndexOutOfBoundsException: Index 62 out of bounds for length 62
     [OF(13)] assms  a_kernel_def
qed

( ring:
  assumes ( UPcgenideal_idealUP assms
proof   ?thesis
  have "\ \ simple_extension K x"
    usingsimple_extension_incl _ assms(1,3[      [Fassms1[ assms3 (4)]java.lang.StringIndexOutOfBoundsException: Index 79 out of bounds for length 79
t ?thesis
    using simple_extension
ub

( ) simple_extension_carrier
carrier "" R
proof -
show <subseteq> simple_extension (carrier R) x"
         subring_is_ring\>simple_extension
next
show
  proof
    fix a assume  proof
      by (induct a rule: simple_extension.induct) (auto Longrightarrow  \<in> simple_extension K' x"
  qed
qed

  java.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 5
assumesjava.lang.NullPointerException
  using mono_simple_extension[OFusingsimple_extension K.simple_extension

lemma (in ring
   "subring'R "K
  using.simple_extension_in_carrier[OF(1)](2-)
  unfolding show byblast

( ring:
  assumes in):
showsKx=\<lambda>p. eval p x) ` carrier (K[X])"
proof
  show "simple_extension K x \ (\p. eval p x) ` carrier (K[X])"
  proof
    fixa  "
    proof (induction rule: simple_extension.induct)
casezero
      have "polynomial K []" and "eval [] x = \"
        unfolding polynomial_def by simp+
      thus ?case
        unfolding univ_poly_carrier
    next
   case(in k2
then java.lang.StringIndexOutOfBoundsException: Index 82 out of bounds for length 82
        by  fixassumek\<in> K" thus "k \<in> simple_extension K x"
      hencejava.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3
         assms2 polynomial_def
      hencefix  "\in>K k
        using eval_append_aux [ _2](,)F(1  java.lang.StringIndexOutOfBoundsException: Index 81 out of bounds for length 81
have p@])java.lang.StringIndexOutOfBoundsException: Index 50 out of bounds for length 50
[ ( <>k2
      then have "local.normalize (p @ [k2]) \ carrier (K [X])"
  using byblast
      ultimately
        unfoldinglemmain) :
    qed
qed
next
(>.evalassumesjava.lang.StringIndexOutOfBoundsException: Index 80 out of bounds for length 80
  proof simple_extensionarrierjava.lang.StringIndexOutOfBoundsException: Index 61 out of bounds for length 61
a"
    then obtain p   proof
using unfolding by auto
    thus       ( a rule.induct simp: assms
    proof
      case0 ?java.lang.StringIndexOutOfBoundsException: Index 23 out of bounds for length 23
        using.zero
    next
      case (Suc
       p'k :"p='@[ ]
        assumes  "
      hence "a = (eval p' x) \ x \ k"
using[ p  ](-)assmsauto
      moreover have "eval p' x \ simple_extension K x"
        using Suc(java.lang.StringIndexOutOfBoundsException: Range [18, 19) out of bounds for length 0
      ultimately " " \<in> simple_extension (carrier R) x" thus "a \<in> carrier R"
using ()p
    qed
  ed
qed

(n domain :
  assumes "subring K proof inductionrule simple_extensioninduct)
using.[OF[OF]
        ring.carrier_is_subring subring  "K\ K'" "x \ K'" shows "simple_extension K x \ K'"
        simple_extension_as_eval_imgpolynomial_def+
    ?case

corollary domain:
"subring KR " \<in> carrier R"
   " K =\ { K'. subring K' R \ K \ K' \ x \ K' }"
  usingby( simp: univ_poly_carrier
        simple_extension_incl[OF         assmslin  by auto
by

) simple_extension_isomorphism
in"
  shows polynomial_incl(]java.lang.NullPointerException
using[eval_ring_homjava.lang.StringIndexOutOfBoundsException: Index 70 out of bounds for length 70
tension_as_eval_img subringE1OF(1)] assms
  unfolding is_ring_iso_def by auto univ_poly_carrier

corollary (in
" K R" "
  showsproof
  using simple_extension_isomorphism[OFsubfieldE[unfolding[OF(1)]  simp
     obtain  p: set

in:
  assumes     ( "length p"arbitrary
        simple_extension
  using simple_extension_isomorphismassumesK\<subseteq> carrier R" "x \<in> carrier R"
.(2[OF]]
  unfolding transcendental_imp_trivial_ker[OFfix "
   java.lang.StringIndexOutOfBoundsException: Index 9 out of bounds for length 9

domain:
  assumesusing.lin(3)unfolding java.lang.StringIndexOutOfBoundsException: Index 61 out of bounds for length 61
subfieldjava.lang.NullPointerException
proof -
       obtain p:" \ carrier (K[X])" "polynomial K p" "eval p x = k1"

    by simp
      corollaryin ):
thenh h:"
      using ring_iso_sym[OF univ_poly_is_ring simple_extension_of_transcendental   "simple_extension Kx \ { K'. subring K' R \ K \ K' \ x \ K' }"
      unfolding is_ring_iso_def by blast
thenHom " <>carrier:simple_extensionKx \" "K[X]" h
      using subring_is_ring[OF simple_extension_is_subring[OF(1)OF) 2]
            univ_poly_is_ring ( domainsimple_extension_isomorphism
      by (auto simp add: ring_hom_ring_def         using    ([  _ K)<.  )java.lang.StringIndexOutOfBoundsException: Index 125 out of bounds for length 125
    have "field (K[X])"
using.[OF(2)OF] h
      unfolding Hom.hom_one is_ring_iso_def
    moreover have "\ field (K[X])"
   "subfieldK "and\<in> carrier R" "(algebraic over K) x"
    ultimately show False by simp
java.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 5
qed

proposition (in " KR "\<in> carrier R" "(transcendental over K) x"
assumes xcarrier
   "ubfield simple_extensionKx R\ (algebraic over K) x"
proof
  assume alg: "(algebraic over K) x"
thenh h  inRuptIrrx)R\<lparr> carrier := simple_extension K x \<rparr>)"
     [OF  is_ring_iso_def
  have rupt_field: "field auto
    using subring_is_ring[OF simple_extension_is_subring[OF subfieldE(1)]]
          rupture_of_Irr
then Hom "Rupt Irr ) R
    using h cring.axioms   "subring R " \<in> carrier R"
    by (auto simp add: ring_hom_ring_def ring_hom_ring_axioms_defproof -
show ( K x)R
usingring_iso_imp_img_field  h](1)OF
              assume "\<not> (algebraic over K)transcendental K) "
    by simp
next
   : " ( K )R"  ( over
    using simple_extension_subfield_imp_algebraicthen h where\inrR
qed

subsection \<open>Link between dimension of K-algebras and algebraic extensions\<close>

lemma ):
  assumes      :  "R\java.lang.StringIndexOutOfBoundsException: Index 100 out of bounds for length 100
  shows "independent K ( by (auto simp add: ring_hom_ring_def ring_hom_ring_axioms_def ring_iso_def)
proof -
  have "\n. n \ degree (Irr K x) \ independent K (exp_base x n)"       fieldring_iso_imp_img_field subfield_iff([OFsimple_ext] h]
  proof -
    fix n show "n \ degree (Irr K x) \ independent K (exp_base x n)"
    proof (    moreover have"<>field([])"
      case (Suc n)
      have "x [^] n \ Span K (exp_base x n)"
      proof (rule ccontr
        assume \notx[] <>Span n)
        then obtain a Ks
          where Ks: "a \ K - { \ }" "set Ks \ K" "length Ks = n" "combine (a # Ks) (exp_base x (Suc n)) = \"
using[OF(1)exp_base_closed (2),ofn]
          by (auto
evalx="
          using combine_eq_eval by (auto simp add: exp_base_def)
        moreover  hen h where"
unfolding polynomial_def(1-2) auto
        ultimately have   rupt_fieldfield( )  "ringR\ carrier := simple_extension K x \)"
using[ subfieldEassms
                IrrE(1          [OF ] assms+
from
      qed
      thus ?case
usingli_Cons assms(2) Suc by (auto simp add: exp_base_def)
    qed
  qed
  thus ?thesis
by simp
qed

lemma (in ring) Span_eq_eval_img:
            simple_extension_in_carrier subfieldE assms(2)]]
  showsjava.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4
( "Span eval_img)
proof
  show "?Span \ ?eval_img"
  proof
    fix u assume "u \ Span K (exp_base x n)"
    then Ks : set
      using Span_eq_combine_set_length_versionjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
      by (auto simp add: exp_base_def)
    hence "u = eval (normalize Ks) x"
      using
"Ks\
-
moreover(Ks
      using normalize_length_leinduct add)
    ultimately ujava.lang.NullPointerException
qed
next
  show "?eval_img \ ?Span"
           aKs
    fix u           Ks a\<> K-{java.lang.StringIndexOutOfBoundsException: Index 134 out of bounds for length 134

      by blast
    hence(( )"
      using combine_eq_eval by auto
    moreover have set_p: "set p \ K"
      using polynomial_incl combine_eq_eval simp:)
     " p \ carrier R"
      using(3[ assms
    moreover have        ultimately "(Kx
using) by auto
    ultimately                 (1[ assms[OF, " "] (3)byjava.lang.StringIndexOutOfBoundsException: Index 88 out of bounds for length 88
using[_[OF()  n]by auto
    moreover have "set ((replicate (n - length p) \) @ p) \ K"
      using ?
    ultimatelyujava.lang.NullPointerException
      using Span_eq_combine_set
  qed
qed

lemma (in domain) Span_exp_base:
  assumes "subfield K R"    ( "?Span =?eval_img")
  shows   proof
  unfolding[OFsubfieldE)OF(1) (2)]
            Span_eq_eval_img   where" Ks\java.lang.StringIndexOutOfBoundsException: Index 99 out of bounds for length 99
proofauto
  interpret "=( Ks "
    using univ_poly_is_principal[OF assms(1)] .
        usingcombine_eq_eval[ _assms subfieldE3[ assms] auto

  fix p assume p: "p \ carrier (K[X])"
   Irr   <in> carrier (K[X])" "Irr K x \<noteq> []"
    using     have normalize
  then qjava.lang.StringIndexOutOfBoundsException: Index 17 out of bounds for length 17
where
      and   proof
    using subfield_long_division_theorem_shell[OFthen  p p:p
hencep=fixassume
    using obtainp(x length
  hence x=  "
    using hom_simps(1) q r unfolding IrrE(4)[OF assms " java.lang.StringIndexOutOfBoundsException: Index 39 out of bounds for length 39
  moreover have     " p \ carrier R"
using((2)  java.lang.StringIndexOutOfBoundsException: Index 31 out of bounds for length 31
ultimately
   "evalpkwherep:p ' [ ]java.lang.StringIndexOutOfBoundsException: Index 43 out of bounds for length 43
    hence(')\ x \ k"
qed

corollary (in domainusing(2[subfieldEassms by auto
  assumesusingusingSuc3unfolding auto
ws" (degreeIrr ) simple_extension )java.lang.StringIndexOutOfBoundsException: Index 63 out of bounds for length 63
  using dimension_independent[OF   shows "Span K (exp_base x (degree (Irr)          simple_extension Suc3)unfoldingby
  by   simple_extension_as_eval_imgin:

lemma):
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
     (12)[Fassms univ_poly_zeroby
proof -
  let ?Us =     where:"

  assume x: "x \ F" then have in_carrier: "x \ carrier R"
    using subringE[OFusing[OF(1  (1)  univ_poly_zero
n where dimension
    using(3)  auto
  have set_Us=r"
using(3,)OF(]by( n) ()
hence" ? subseteq> carrier R"
    using subringE(1)[OF assms2)] by auto
  moreover have "
    using independent_length_le_dimension[OF   "subringK ""x\java.lang.StringIndexOutOfBoundsException: Index 43 out of bounds for length 43
  ultimately
  obtain Ks        simple_extension_incl subringE1OF(1)] assms java.lang.StringIndexOutOfBoundsException: Index 97 out of bounds for length 97
    usingshows)  (K[X)R\lambda. )
have java.lang.NullPointerException
    using[OF(1)OF() (2)]
  hence normalize
    using  ( domain:
moreover" Ksjava.lang.StringIndexOutOfBoundsException: Index 84 out of bounds for length 84
by Ks, list,
                    metis all_not_in_conv listunfolding[ assms by
  hence "normalize Ks \ []"
    using Ks(1,4) by
   havenormalize
    using normalize_gives_polynomial[OF Ks(3)] unfolding univ_poly_def by auto
  ultimately ?thesis
    using algebraicI by auto
qed

( ) :
  java.lang.StringIndexOutOfBoundsException: Range [0, 9) out of bounds for length 0
  showsK) K(  "
  using dimI[OF assms(1) dimension_simple_extension[OF assms]] .

corollary (in domain) finite_dimension_simple_extension:
  assumes    (    dvd   java.lang.StringIndexOutOfBoundsException: Index 31 out of bounds for length 31
  shows
  using finite_dimensionI[ ( domain :
_java.lang.StringIndexOutOfBoundsException: Range [72, 71) out of bounds for length 89
simple_extension_memsubfieldE
  bya

\<open>Finite Extensions\<close>

( ring:
  unfolding Hom java.lang.StringIndexOutOfBoundsException: Index 48 out of bounds for length 48
proof -
havejava.lang.StringIndexOutOfBoundsException: Index 109 out of bounds for length 109
  proof -
    fix K' (in domain simple_extension_is_subfield:
      using ring.finite_extension.simps[OF    "subfield ( K x
            simple_extension_consistent[OF alg ( )
java.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 5
  thus ?thesis by blast
qed

lemma (in ring) mono_finite_extension] by+
"
  using mono_simple_extension assms by (induct xs) (auto)

lemma (in ring) finite_extension_carrier:
  assumesby( simp: ring_hom_ring_def ring_iso_def
  using assms simple_extension_carrier by (induct .ring_iso_imp_img_field  h]          simple_extension_in_carrier[OF subfieldE(3)[OF assms(1

lemma (in ring) finite_extension_in_carrier:
assumesjava.lang.StringIndexOutOfBoundsException: Index 122 out of bounds for length 122
   assms byinductauto

lemma (in ring( ) :
  K' R Klemma( ring)finite_dimension_imp_algebraic:
  java.lang.StringIndexOutOfBoundsException: Range [41, 40) out of bounds for length 84
  unfolding finite_extension_consistent[OF assms(1)       (Suc

lemma( ring      roofule)
           java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
hows  <   x xs
  using[OF finite_extension_in_carrier[OF(1,3)] assms]by  obtain wheren n: dimensionjava.lang.StringIndexOutOfBoundsException: Index 37 out of bounds for length 37

lemma (in         "eval (a# ) x = \"
  assumes "K \ carrier R" and "set xs \ carrier R" shows "K \ finite_extension K xs"
  using         moreoverhaveset_Us " (?Usn \ F"

lemma (in ring  henceset        have"degree (IrrKx)<>n"
  assumes "K \ carrier R" and "x \ carrier R" "set xs \ carrier R"
  shows "finite_extension K (x # xs) = usingsubringE1[OF assms(2) java.lang.StringIndexOutOfBoundsException: Index 42 out of bounds for length 42
on_in_carrierOFIrrE)OF _ _  Irr_minimalassmsofa #"] Ks3 by java.lang.StringIndexOutOfBoundsException: Index 88 out of bounds for length 88

lemma         \<open>Suc n \<le> degree (Irr K x)\<close> and this show False by simp
assumes K R" " xs
  using assms simple_extension_is_subring by (induct xs) (auto)         independentli_Cons(2) Suc( Usn =\<zero>" "set Ks \<subseteq> K" "set Ks \<noteq> { \<zero> }"

corollary (indomain)finite_extension_mem
  assumessubring  java.lang.StringIndexOutOfBoundsException: Index 32 out of bounds for length 32
shows xs
proof (induct xs)
  case Nil
  then     (is "?Span?")
next
caseCons a xs)
  from Cons(2) have a: "a \ carrier R" and xs: "set xs \ carrier R" by auto
  show ?case
proof
fix"\in (#)java.lang.StringIndexOutOfBoundsException: Index 39 out of bounds for length 39
then=|"\
    then show "x \ finite_extension K (a # xs)"
    proof cases
      case 1
      with a have "x \ carrier R" by simp
      with xs have "x \ finite_extension K (x # xs)"
        usingsimple_extension_mem[OF [OF subringby simp
      with 1 show ?thesis by simp
    next
      case    moreover "normalizeKs carrier (K[X])"
*:\<in> finite_extension K xs" by auto
  havefinite_extension
        using])
          [OF()]  byjava.lang.StringIndexOutOfBoundsException: Index 78 out of bounds for length 78
    qed
  java.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 5
qed uassume u

corollary (in domain) finite_extension_minimal:
  assumes "subring K R" "set xs \ carrier R"
showsfinite_extensionxsjava.lang.NullPointerException
  using      by blast
        finite_extension_incl[OF subringE(1)[OF assms(1)] assms(2)]        combine_eq_eval auto
  by blast

( )finite_extension_same_set
  assumes  shows(dim over K) (simple_extension       polynomial_inclofp p()unfolding by auto
  shows "finite_extension K xs = finite_extension K ys"
  finite_extension_minimal assms]assms)byauto

text
proposition (       p
   ultimately(( length
  shows      usingcombine_prepend_replicate[OF exp_base_closed[OF assms) of n]]by
  using algebraic_mono assms
  by (induct xs) (auto, metis finite_extension.simps finite_extension_incl subring_props(1))

proposition (in domain) finite_extension_finite_dimension show " ?pan"
  assumes "subfield K R" "set xs \ carrier R"
   "(\x. x \ set xs \ (algebraic over K) x) \ finite_dimension K (finite_extension K xs)"
java.lang.StringIndexOutOfBoundsException: Index 135 out of bounds for length 135
proof -
  show "finite_dimension K (finite_extension java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
    using   subfield ""x \<in> carrier R" "(algebraic over K) x"
finite_extension_is_subring([ (1]2]
          finite_extension_mem[OF subfieldEunfolding[OF subfieldE[ assms() assms(2)]
next
show"\Andx. x set xs \ (algebraic over K) x) \ finite_dimension K (finite_extension K xs)"   L>(  )x <Longrightarrow> finite_dimension K (finite_extension K xs)"
    using assms(2)
  proof induct, simpaddfinite_dimensionI
    case (subsection
    hence "finite_dimension K (finite_extension K xs)"
      by auto
    moreover have "(algebraic over (finite_extension K xs)) x"
singOF[3 assms]Consauto
    moreover have "subfield (finite_extension K xs) R"
      using[OF(1]Cons java.lang.StringIndexOutOfBoundsException: Index 71 out of bounds for length 71

      using java.lang.StringIndexOutOfBoundsException: Index 9 out of bounds for length 9
       .finite_extension[OF subring_is_ringassms
qed
qed

corollary     subfield_long_division_theorem_shell assms1pIrr unfolding  univ_poly_zero by auto
  assumes "subfield K R" "set 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 assmsjava.lang.StringIndexOutOfBoundsException: Index 50 out of bounds for length 50
        [OFsubfieldE[ assms()] assms]
        finite_extension_finite_dimension(1)[OF assms(1- moreover " r < length ( K x)"

corollary
  assumes   "evalp
  shows "y \ simple_extension K x \ (algebraic over K) y"
  using finite_extesion_mem_imp_algebraic[OF assms(1), of " using r by auto

subsection \<open>Arithmetic of algebraic numbers\<close>

text \<open>We show that the set of algebraic numbers of a field
      over  asubfield

lemma (in) subfield_of_algebraics
    by( add: exp_base_def
proof -
  let ?set_of_algebraics = "{ x \ carrier R. (algebraic over K) x }"

  show ?thesis
  proof (rule subfieldI'[OFlemma(inring) finite_dimension_imp_algebraic:
show"?set_of_algebraics \ carrier R" and "\ \ ?set_of_algebraics"
using[OF _subringE]subfieldE)OF(1)] by auto
  next
    fix x y assume x: "x \ ?set_of_algebraics" and y: "y \ ?set_of_algebraics"
    have "\ x \ simple_extension K x"
      using   x "x\ F" then have in_carrier: "x \ carrier R"
            [OFsubfieldE (1) x byauto
    thus "\ x \ ?set_of_algebraics"
using[ ]   java.lang.StringIndexOutOfBoundsException: Index 65 out of bounds for length 65

   have"ependentK(Us n)java.lang.StringIndexOutOfBoundsException: Index 37 out of bounds for length 37
obtain where:" Ks = Sucn"combineKs(Us)  \<zero>" "set Ks \<subseteq> K" "set Ks \<noteq> { \<zero> }"
    using[OFassms)  "?Us n] 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
  java.lang.StringIndexOutOfBoundsException: Index 6 out of bounds for length 6
fix :
    have "inv z \ simple_extension K z"
ield_m_invin mono_finite_extension
            simple_extension_is_subfield[OF, of java.lang.StringIndexOutOfBoundsException: Index 56 out of bounds for length 56
"xs .3 subset_singletonD
thus  inductjava.lang.StringIndexOutOfBoundsException: Index 60 out of bounds for length 60
simple_extesion_mem_imp_algebraicassms z by
  qed
qed

end

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