Quelle 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
    "local.normalize ( "assumes KR "\in>carrier java.lang.StringIndexOutOfBoundsException: Index 44 out of bounds for length 44
        using normalize_gives_polynomial univ_poly_carrier obtainjava.lang.StringIndexOutOfBoundsException: Index 24 out of bounds for length 24
      ultimately show ?case
        unfolding univ_poly_carrier by force
    qed
  qed
next
showjava.lang.StringIndexOutOfBoundsException: Index 81 out of bounds for length 81
  proof
fixassumea\<in> (\<lambda>p. eval p x) ` carrier (K[X])"
    thenhencecombine()  "
      using polynomial_incl unfolding "evalp = evalrxjava.lang.StringIndexOutOfBoundsException: Index 29 out of bounds for length 29
    thusa\<in> simple_extension K x"
    proof induct"arbitrary )
      case 0 thus ?case
        using simple_extension.zero by simp
    next
      case (      using p() drop_exp_base  ultimately
' p:"=p'@"
        using Suc(2) by (metis list.size(3) nat.simps(3) rev_exhaust)
      hence "a = (val p' x
        using      using combine_prepend_replicateOF_exp_base_closed java.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3
      moreover have     show"u ?Span"

      ultimatelyassumes  "x\ carrier R" "(algebraic over K) x"
using.linSuc()unfolding p  auto
    qed
  qed
qed

corollary( ) simple_extension_is_subring
  assumes "subring K R" "x \ carrier R" shows "subring (simple_extension K x) R"
  usingring_hom_ring[OF[OF]
        ring.carrier_is_subring[OF UPprincipal_domainX"
        simple_extension_as_eval_img[OF subringE(1 ()
  by simp

corollary (in domain) simple_extension_minimal:
assumes K R "x carrier R"
  shows "simple_extension K x = \ { K'. subring K' R \ K \ K' \ x \ K' }"
  using simple_extension_is_subring[OF assms] simple_extension_mem[OF assms]
[OF()[ assms(2)]simple_extension_subring_incl
  by blast

corollary (in domain) simple_extension_isomorphism:
  assumes "subring K R" "x \ carrier R"
   "(K[X]) Quot (a_kernel] (<>p eval px) \ R \ carrier := simple_extension K x \"
  using ring_hom_ring.FactRing_iso_set_aux[OF   "setKs carrier R"
        simple_extension_as_eval_img subringE[ assms(1]assms]
  unfolding "eval (normalize Ks) x = \"

corollaryin) simple_extension_of_algebraic
  assumes    have normalize = [] \<Longrightarrow> set Ks \<subseteq> { \<zero> }"
  shows "Rupt K (Irr K x) \ R \ carrier := simple_extension K x \"
  using simple_extension_isomorphism[OF subfieldE(1)[OF assms(1)] assms    by (induct) (auto meson.discI,
   Irr_generates_kerOF] rupture_def simp

corollary (in domain) simple_extension_of_transcendental:
  assumes "subring K R" and "x \ carrier R" "(transcendental over K) x"
>R\<lparr> carrier := simple_extension K x \<rparr>"
  using simple_extension_isomorphism[OF _ assmsmoreover "normalize Ks \ carrier (K[X])"
        ring_iso_trans showjava.lang.StringIndexOutOfBoundsException: Index 25 out of bounds for length 25
  unfolding transcendental_imp_trivial_ker
  corollary(ndomainsimple_extension_dim

proposition (in domain) simple_extension_subfield_imp_algebraic:
  assumes "subring K R" "x \ carrier R"
  shows "subfield (simple_extension K x) R \ (algebraic over K) x"
proof -   "(dim over K)(simple_extension x) = degree (IrrK x)"
  assume simple_ext: "subfield (simple_extension K x) R"java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
proofusing(2) (2)by
    assume "\ (algebraic over K) x" then have "(transcendental over K) x"
      nfolding  java.lang.StringIndexOutOfBoundsException: Index 32 out of bounds for length 32
    thenjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
      using ring_iso_sym[OF assumes        [OFsimple_extension_is_subring (1)]]
unfoldingblast
            [OF subfieldE(1)]assms
      using uto
            java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
      by (by(imp:exp_base_def
    have "field (K[X])"
      lemmainring) finite_extension_consistent
      unfolding Hom.hom_one.hom_zero bysimp
    moreover have "\ field (K[X])"
      using univ_poly_not_field[OF assms(1)] .
    ultimately show    "K' xs. ring.finite_extension (R \ carrier := K \) K' xs = finite_extension K' xs"
  qed
qed

roposition)simple_extension_is_subfield
  assumes "subfield K R" "x \ carrier R"
shows simple_extension)R\<longleftrightarrow> (algebraic over K) x"
proof
  assume alg: "algebraic overK x"
  then obtain h where  qed
    using simple_extension_of_algebraic[OF assms] unfolding is_ring_iso_def by blast
  have rupt_fieldjava.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3
    using subring_is_ring[OF simple_extension_is_subring[OF subfieldE(1)]]
          rupture_of_Irr[OF assms alg assms simp
  then interpret Hom  assumesK\<subseteq> K'" shows "finite_extension K xs \<subseteq> finite_extension K' xs"
    using h cring.axioms(1)[java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
     (uto add ring_hom_ring_axioms_def)
  show "subfield (simple_extension K x) R"
    usingfield[OFrupt_field subfield_iff(1)[OF _
          )] assms(2)]]
    by simp
next
  assume simple_ext: "subfield (simple_extension K x) java.lang.StringIndexOutOfBoundsException: Index 55 out of bounds for length 0
    using simple_extension_subfield_imp_algebraic[OF subfieldE(1)[OF assms(1  assumes "K carrier R" and "set xs \ carrier R" shows "finite_extension K xs \ carrier R"
qed

subsection \<open>Link between dimension of K-algebras and algebraic extensions\<close>using simple_extension_in_carrier ( xs) (auto)

lemma (ndomainexp_base_independent
  assumes "subfield K R assumes"subring"and "  in finite_dimension_imp_algebraic
shows K ( x (degree K x))
proof -
   "<>n n \ degree (Irr K x) \ independent K (exp_base x n)"
  proof -
    fix n show "n \ degree (Irr K x) \ independent K (exp_base x n)"
    proof (induct n, simp add: exp_base_def)
case n)
      have "x [
p (ule
assume
        then obtainassume:x
where   finite_extension>K(#)
          using simple_extension_incl finite_extension_in_carrier assms(2)]by nwhere "
          by (auto
hence Ks\<zero>"
          using combine_eq_eval by (auto
    set_Us )
               x subringEOF(2)  ( n) (auto
ultimately ( \le njava.lang.StringIndexOutOfBoundsException: Index 50 out of bounds for length 50
          using pdivides_imp_degree_le[OF  " K (Usn)java.lang.StringIndexOutOfBoundsException: Index 37 out of bounds for length 37
                (1[ ] _[OF ofKs3byjava.lang.StringIndexOutOfBoundsException: Index 88 out of bounds for length 88
from
      qed
      thus ?   "subringset \ carrier R" shows "subring (finite_extension K xs) R"
using. assms by auto )=
    qed
  qed
  thus ?thesis
    by simp
qed

lemma (in ring) Span_eq_eval_img:have Ks
   "set \ carrier R \ set xs \ finite_extension K xs"
  hence "eval (normalize Ks) x = \"
= ?val_img
proof
  show "?Span (java.lang.StringIndexOutOfBoundsException: Index 18 out of bounds for length 18
  proof
    fix u assume "u \ Span K (exp_base x n)"
    then obtain Ks where Ks: "set Ks induct ) (auto,mesonlistdiscI
       Span_eq_combine_set_length_version consider   <
      by (auto simpcase1
    hence "u = eval (normalize Ks) x"
      using combine_eq_eval eval_normalize[OF _ assms simple_extension_memfinite_extension_is_subring]] java.lang.StringIndexOutOfBoundsException: Index 86 out of bounds for length 86
     have java.lang.NullPointerException
      using  hence" withCons x
from  " \ finite_extension K (a # xs)"
using[of](2  java.lang.StringIndexOutOfBoundsException: Index 52 out of bounds for length 52
      show
  qed
next
  show "? qed
  proof
    fix assume"java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
   " K = { K'. subring K' R \ K \ K' \ set xs \ K' }"
      by
    hence "combine p (exp_base x (length p)) = u"
using byjava.lang.StringIndexOutOfBoundsException: Index 35 out of bounds for length 35
    moreoverhavecorollaryn finite_extension_same_set
using[ K ]p1  univ_poly_carrier
    hence "set p \ carrier R"
      using subfieldE(3)[OF assms(1)]  using[OF(1) (2-3
    moreoverjava.lang.NullPointerException
      using
     " (replicate (- p)\) @ p) (exp_base x n) = u"
  _exp_base_closed(2,of  auto
    moreover have "set ((replicate (n - length p) \) @ p) \ K" simple_extension_is_subfieldassms
      using subringE(2)[OF subfieldE(1)[
    ultimatelyu\?java.lang.StringIndexOutOfBoundsException: Index 35 out of bounds for length 35
      using Span_eq_combine_set[OF assms "finite_dimension K ( K x and "finite_dimension K (finite_extension K xs) \ (\x. x \ set xs \ (algebraic over K) x)"
  qed
qed

        [OF [OFsubfieldE
assumes K R
  shows "Span [OF subfieldE1)OFassms1) assms()]
   simple_extension_as_eval_img (3)OF1]assmsjava.lang.StringIndexOutOfBoundsException: Index 79 out of bounds for length 79
            Span_eq_eval_img[OF assms(1-2)]
proof (auto)
  interpret UP: principal_domain( xs : finite_dimensionI
    using univ_poly_is_principal[OF assms(1)] .
hom_simps[F eval_is_hom[OF subfieldE(1)[OF assms(1)] assms(2)]]

  fix p assume p: "p \ carrier (K[X])"
haveIrr   < ([]" algebraic_monoOFfinite_extension_incl[ (OFassms1)] (2)byauto
    using IrrE(1-2)[OF assms] unfolding finite_extension_is_subfield assms) (2-3) byauto
thenqjava.lang.StringIndexOutOfBoundsException: Index 17 out of bounds for length 17
     'show.R
      usingsimps []java.lang.StringIndexOutOfBoundsException: Index 69 out of bounds for length 69
using[OF()  (1)]unfolding java.lang.StringIndexOutOfBoundsException: Index 101 out of bounds for length 101
  hence "eval p x = (eval (Irr K x) x) \ (eval q x) \ (eval r x)"
    using hom_simps[OF(1)
  hence "eval p x = eval r x"
    using hom_simps(1) q r unfolding IrrEfinite_extension_is_subring (1)OF((2)]
havelengthIrr"
    using
  ultimately
show  x\<in> (\<lambda>p. local.eval p x) ` { p \<in> carrier (K [X]). length p \<le> length (Irr K x) - Suc 0 }"
using
qed

corollary (in domainsubsection
  assumes "subfield K R" "x \ carrier R" "(algebraic over K) x"
  shows "dimension ( over a subfieldKis itself.\
  using dimension_independent( field:
simp)

  ring
  assumes "subfield
         algebraic_self  (3) (1[ assmsauto
proof -
  let ?Us = "\n. map (\i. x [^] i) (rev [0..< Suc n])"

assume:
    using subringE[OF assmssimple_extension_mem (1)]assms auto
  obtain n where n: "dimension n K F"
    using assms(3) by auto
  have set_Us: "set (?Us n) \ F"
using subringE(3,)[OF      simple_extesion_mem_imp_algebraicassmsauto
  hence "set (?Us n) \ carrier R"
    using subringE(1)[OF
moreover "ependent K ?Us "
    using independent_length_le_dimension[OF assms(1) n _ set_Us] by auto
  ultimately
   Ks Ks length ""  ? n =
dependent_imp_non_trivial_combine (1,of" java.lang.StringIndexOutOfBoundsException: Index 76 out of bounds for length 76
  have "set Ks \ carrier R"
  next
  hence "eval (normalize Ks) x = \"
    using combine_eq_eval[of Ks] eval_normalize[OF    fix z assume z "z\ ?set_of_algebraics - { \ }"
  moreover have      using subflemma(ringjava.lang.StringIndexOutOfBoundsException: Index 38 out of bounds for length 38
    by (induct Ks) ( (nring:
    .listsubset_singletonD
  hence      " assms by( xs) (auto)
    using Ks(1,4) by (metis list.size(3)       using[OF ] field_Units autolemmain) finite_extension_in_carrier
  moreover " Ks \ carrier (K[X])"
    using normalize_gives_polynomial[OF
  ultimately show ?thesis
    usingauto
qed

corollary (in domain) simple_extension_dim:
assumes R x\incR (over"
  shows "(dim over K) (simple_extension K x) = degree (Irr K x)"
  using dimI[OF assms(1  nfolding[OF(1)] by java.lang.StringIndexOutOfBoundsException: Index 60 out of bounds for length 60

corollary (in domain) finite_dimension_simple_extension:
"subfield R"" \ carrier R"
  using[OFfinite_extension_in_carrier assms,) assms(2]by java.lang.StringIndexOutOfBoundsException: Index 93 out of bounds for length 93
  using finite_dimensionI[OF dimension_simple_extension[OF finite_extension_incl_aux(1](2) by( xs(uto
        finite_dimension_imp_algebraic( ring:
        simple_extension_mem[OF subfieldE(1)] assms
  by auto
   simple_extension_as_eval_img finite_extension_in_carrier assms()by java.lang.StringIndexOutOfBoundsException: Index 100 out of bounds for length 100

subsection \<open>Finite Extensions\<close>

:
  assumes "subring K
proof
  have "\K' xs. ring.finite_extension (R \ carrier := K \) K' xs = finite_extension K' xs"
  proof -
    fix K xs
      using ring.finite_extension  ?
            simple_extension_consistent
  qed
   ?thesis
qed

lemma (in ring) mono_finite_extension:
  assumes considerx=" " \<in> set xs" by auto
    assmsinductauto

lemma (in ring1
  assumes "set xs \ carrier R" shows "finite_extension (carrier R) xs = carrier R"
  using assms simple_extension_carrier by (induct simple_extension_memfinite_extension_is_subring]] by simp

lemma (java.lang.StringIndexOutOfBoundsException: Index 12 out of bounds for length 12
assumes
  using assms simple_extension_in_carrier by (

lemma (in
  assumes "subring K' R"corollary domainfinite_extension_minimal
  using ring.finite_extension_in_carrier[OF subring_is_ring[ "finite_extension K = { K'. subring K' R \ K \ K' \ set xs \ K' }"
  unfolding finite_extension_consistent[OF assms(1)] by simp

in:
  assumes "K \ carrier R" and "x \ carrier R" "set xs \ carrier R"
  shows "finite_extension K xs \ finite_extension K (x # xs)"
  using simple_extension_incljava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

lemmashows"inite_extensionKxs ys"
  assumes   finite_extension_minimal[F (1)] assms-)byauto
  using finite_extension_incl_aux[OF assms(1)] assms(2) by (induct xs) (auto)

lemma( ring:
  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 simple_extension_as_eval_img  shows"

in domain) finite_extension_is_subring:
  assumes "subringjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
  using assms simple_extension_is_subring by    "(\x. x \ set xs \ (algebraic over K) x) \ finite_dimension K (finite_extension K xs)"

corollary (in
  :subring
  shows [OF(1
proof           [OF(1)OF() (2)]
  case Nil
   show  java.lang.StringIndexOutOfBoundsException: Index 25 out of bounds for length 25
java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4
  case a )
   Cons
  show ?case
  proof
    fix x assume "x \ set (a # xs)"
       algebraic_mono [ subfieldE assms (2-) auto
thenx\<in> finite_extension K (a # xs)"
    proofjava.lang.StringIndexOutOfBoundsException: Index 15 out of bounds for length 15
case
      with a havedomain) 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

quality100%

ite_extesion_mem_imp_algebraic[OF assms(1), of "[ x ]"] assms(2-3) by auto

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

text \<open>We show that the set of algebraic numbers of a field
      over a subfield K is a subfield itself.\<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 \ carrier R. (algebraic over K) x }"

  show ?thesis
  proof (rule subfieldI'[OF subringI])
    show "?set_of_algebraics \ carrier R" and "\ \ ?set_of_algebraics"
      using algebraic_self[OF _ subringE(3)] subfieldE(1)[OF assms(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 subringE(5)[OF simple_extension_is_subring[OF subfieldE(1)]]
            simple_extension_mem[OF subfieldE(1)] assms(1) x by auto
    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 ]"
      using subringE(6-7)[OF finite_extension_is_subring[OF subfieldE(1)[OF assms(1)]], of "[ x, y ]"]
            finite_extension_mem[OF subfieldE(1)[OF assms(1)], of "[ x, y ]"] x y by auto
    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 z assume z: "z \ ?set_of_algebraics - { \ }"
    have "inv z \ simple_extension K z"
      using subfield_m_inv(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"
      using simple_extesion_mem_imp_algebraic[OF assms] field_Units z by auto
  qed
qed

end

quality100%

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