Impressum Finite_Extensions.thy Sprache: Isabelle

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

theory    AuthorFinite_Extensions
   Polynomials

begin

section \<open>Finite Extensions\<close>

subsection \<open>Definitions\<close>

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

abbreviation (in ring) algebraic :: "'a set \ 'a \ bool"
  where "algebraic K xjava.lang.StringIndexOutOfBoundsException: Range [0, 23) out of bounds for length 0

definition (in java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
whereIrrx=THE <  KX)java.lang.StringIndexOutOfBoundsException: Index 130 out of bounds for length 130

inductive_set (in ring) simple_extension :: "'a set \ 'a \ 'a set"
  forKand in)  :: ' 'a \ 'a list"
    zerowhere  =( .p\<in> carrier (K[X]) \<and> pirreducible K p \<and> eval p x = \<zero> \<and> lead_coeff p = \<one>)"algebraic
    where x=(pjava.lang.NullPointerException

funfor  x java.lang.StringIndexOutOfBoundsException: Index 19 out of bounds for length 19
where=

subsectionin  :: ' Rightarrow alist\ 'a set"

lemma " Kxs (\x K'. simple_extension K' x) xs K"
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
unfolding ringOF assms
            univ_poly_consistent[OF assms unfolding infinite_extension'set\ 'a list \ 'a set"

lemma (in ring) algebraic_consistent:
  assumes "subring
  unfoldingover_def[OF ] ..

lemma in)eval_transcendental
assumestranscendental)x p <>carrier""    <>"shows "p =["
proof "( assumessubring "   transcendental
   "[] \ carrier (K[X])" and "eval [] x = \"
  have <in> carrier (K[X])" and "eval [] x = \<zero>"in) algebraic_consistent
  thus
    using assms unfolding over_def transcendental_def   unfoldingover_def transcendental_consistent    byauto adduniv_poly_defjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
qed

lemma in) transcendental_imp_trivial_ker
   proof
  usingeval_transcendental shows([X)R\<

lemma (in ring) non_trivial_ker_imp_algebraic:
  shows "a_kernel (K[X]) R java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
  using unfoldingby auto

lemma   usingring_hom_ring ?thesis  unfolding over_defby ( add)

   ( over)x\<
  using.trivial_ker_imp_inj eval_ring_hom assms
  unfolding transcendental_def over_def by (simp add: univ_poly_zero)

lemma (in "algebraic K)x\java.lang.StringIndexOutOfBoundsException: Index 105 out of bounds for length 105
  singover_def
  showsassumes) :
     p  "p\ carrier (K[X])" "p \ []" "eval p x = \"

lemma java.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 7
  assumesusing.trivial_ker_imp_inj[OF OF' by java.lang.StringIndexOutOfBoundsException: Index 82 out of bounds for length 82
     trivial_ker_imp_transcendental "
proof -
  have "[] \ a_kernel (K[X]) R (\p. eval p x)"
    unfolding' by auto
thenobtain "p carrier (K[X])" "p \ []" "eval p x = \"
     algebraic_imp_non_trivial_kerOF ] unfolding a_kernel_defoverx
thus KR and <in> carrier R" "(algebraic over K) x" ( overx

lemma (in ?thesisusing   obtainsp
java.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3
  using assms non_trivial_ker_imp_algebraicusing[ assms(2)unfolding ' java.lang.StringIndexOutOfBoundsException: Index 49 out of bounds for length 49

lemma in) transcendental_mono
assumes
proof -
  have " [ var_closed()OFassms] unfoldingvar_def
     assms polynomial_def
  thus ?   assms non_trivial_ker_imp_algebraic a_kernel_defbyauto
     assmsunfolding transcendental_def by( inj_on_subsetrule "[\, \ k ]"])
qed

corollaryinalgebraic_mono
  assumes
   subringE java.lang.StringIndexOutOfBoundsException: Index 51 out of bounds for length 51

lemma(  :
  assumes   assumes K R"
  corollaryin) algebraic_mono

lemma (in domain   transcendental_mono[ assms
  assumessubring andjava.lang.StringIndexOutOfBoundsException: Index 68 out of bounds for length 68
  thus
u a_kernel_defjava.lang.StringIndexOutOfBoundsException: Index 36 out of bounds for length 36
     ( )
  have domain  " K R x\in> carrier R"" K xjava.lang.StringIndexOutOfBoundsException: Index 71 out of bounds for length 71
(OF(by
  thus "eval [ \, \ k ] k = \"
    by(utoalgebra
qed

lemma(n ) ker_diff_carrier
  assumes subring"
   "a_kernel([]) java.lang.StringIndexOutOfBoundsException: Index 74 out of bounds for length 74
proof-
  have "eval [ \ ] x \ \" and "[ \ ] \ carrier (K[X])"
    using subringE  thus
  thustjava.lang.StringIndexOutOfBoundsException: Index 14 out of bounds for length 14
    unfolding' by by(, algebra)
qed

subsection[OF_assmsshows([) <>.px)\<noteq> carrier (K[X])"

lemma( domain minimal_polynomial_is_unique 3[ java.lang.StringIndexOutOfBoundsException: Index 78 out of bounds for length 78
  subsection
  shows "\!p \ carrier (K[X]). pirreducible K p \ eval p x = \ \ lead_coeff p = \"
    (is "\!p. ?minimal_poly p")
proof
      have"q. ?minimal_poly q \ q = p"
    using univ_poly_is_principal assms]java.lang.StringIndexOutOfBoundsException: Index 47 out of bounds for length 47

  let      usingunfolding' by java.lang.StringIndexOutOfBoundsException: Range [45, 46) out of bounds for length 45
                           UP q p byjava.lang.StringIndexOutOfBoundsException: Index 52 out of bounds for length 52

  obtain  qed
     exists_unique_pirreducible_gen assms[ _ assms
          lemmain):
ker_diff_carrier[ assms)  auto
  hence "? using exists_unique_p " (IrrK)=\<" " ( K x) x=\zero"
    using UP.cgenideal_self p unfolding a_kernel_defalgebraic_imp_non_trivial_ker assms
   have\Andq \Longrightarrowq="
  proof
     q  q "minimal_poly
    then " > PIdl\<^bsub>K[X]\<^esub> p"
u p unfolding' by auto
    hence "p \\<^bsub>K[X]\<^esub> q"
[  KR  xjava.lang.StringIndexOutOfBoundsException: Index 71 out of bounds for length 71
       ker(] <> px  \<^bsub>K[X]\<^esub> q"
.associated_iff_same_ideal simp
    thus "q = p"
usingqbyjava.lang.StringIndexOutOfBoundsException: Index 28 out of bounds for length 28
qed
  ultimately   java.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 5

lemma java.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3
  assumes " ssumes" K R" "x \<in> carrier R" "(algebraic over K) x"
  shows          .[OFuniv_poly_is_cring assms]
    and "lead_coeff (Irr K x) = \" and "eval (Irr K x) x = \"
  using ker

lemma( domain) Irr_generates_ker:
java.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3
shows([X]  java.lang.StringIndexOutOfBoundsException: Index 87 out of bounds for length 87
proof -
  obtainusing[OF assms1 eval_ring_hom _ assms2]
where q\<in> carrier (K[X])" "pirreducible K q"
and: "a_kernel([]R(\p. eval p x) = PIdl\<^bsub>K[X]\<^esub> q"
       "Irr K \java.lang.StringIndexOutOfBoundsException: Index 49 out of bounds for length 49
          [OF3]
          ker_diff_carrier] subfieldE K.lin
  by
    using
  thus show? by blast
    using cgenideal_pirreducible    by(meson.cgenideal_ideal.cgenideal_minimal(4))
          .associated_iff_same_ideal univ_poly_is_cring(1)OF
    unfolding ker
    ysimp
qed

lemma (in domain) Irr_minimal  proof(induct: simple_extensioninduct
  assumes "subfield K R" andassumes  "and" qed
p\in  []""  java.lang.StringIndexOutOfBoundsException: Range [43, 42) out of bounds for length 82
proof -
   UP
    using univ_poly_is_principal kassume" K" thus "k \ simple_extension K x"

  have "p \ PIdl\<^bsub>K[X]\<^esub> (Irr K x)"
usingIrr_generates_kerOF assms-(4-5)unfolding' by auto
  hence
lemmain) simple_extension_mem
    by(eson.cgenideal_ideal .cgenideal_minimal(4))
  thusjava.lang.StringIndexOutOfBoundsException: Index 14 out of bounds for length 14
unfolding[ 1 ()[Fassms(-]assms].
qed

lemma (in domain) rupture_of_Irr:
  assumes "subfield K R" and "x \ carrier R" "(algebraic over K) x" shows "field (Rupt K (Irr K x))"
usingjava.lang.StringIndexOutOfBoundsException: Index 19 out of bounds for length 14

section

lemmalemmainsimple_extension_carrier
assumesK"showsringsimple_extension( carrier := K \) = simple_extension"
proof-
     "carrier R\java.lang.StringIndexOutOfBoundsException: Index 61 out of bounds for length 61
usinge_extension subseteq' "

  havejava.lang.StringIndexOutOfBoundsException: Range [0, 7) out of bounds for length 4
java.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 7
>a
      by (induction rule: K.simple_extension
qed
  moreover
have"<>K .simple_extensionK
  proof
    fix K' x "K carrier R" and "x \ carrier R" shows "simple_extension K x \ carrier R"
       K..zero.lin
      by (induction rule: simple_extension
  qed
  ultimately showassumes K "and "K\<subseteq> K'" "x \<in> K'" shows "simple_extension K x \<subseteq> K'"by( rule: simple_extensioninduct)+
lemin) simple_extension_as_eval_img

lemma( ring mono_simple_extension
  assumes    "simple_extension K =(\p. eval p x) ` carrier (K[X])"
java.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 5
  fix a assumea \<in> simple_extension K x" thus "a \<in> (\<lambda>p. eval p x) ` carrier (K[X])"
  proof (induct a rule      case java.lang.StringIndexOutOfBoundsException: Index 15 out of bounds for length 15
unfolding by force
  qed
qed ( k1)

lemma obtainp where p: "p \ carrier (K[X])" "polynomial K p" "eval p x = k1"
  assumes
proof
   k  " \ K" thus "k \ simple_extension K x"
    using simple_extension.lin[OF simple_extension.zero, of k K x]  qed
qed

lemma (in ring   "K \ carrier R" and "x \ carrier R" shows "K \ simple_extension K x"
  assumes        using(1) lin() unfolding by auto
proof
  have "\ \ simple_extension K x"
usingOFassmssubringE[assmsbyjava.lang.StringIndexOutOfBoundsException: Index 81 out of bounds for length 81
    univ_poly_carrier
    using simple_extension show
qedlemma( ring simple_extension_carrier

    qed
  assumes
proof
  show  show"\x<in> carrier R" shows "simple_extension (carrier R) x = carrier R"
    using    "carrierR\ simple_extension (carrier R) "
next
  showsimple_extension(  x <subseteq> carrier R"
  proof
fixassumea\<in> simple_extension (carrier R) x" thus "a \<in> carrier R"
      by              simple_extension3
  q
qed

lemma
  corollaryin domain)simple_extension_is_subring
proof :simple_extension

lemma (in   ring_hom_ringimg_is_subring eval_ring_hom assms
  assumes" K' R"and \<subseteq> K'" "x \<in> K'" shows "simple_extension K x \<subseteq> K'"       "polynomialK ] eval [ "
.java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
:)

lemmau . bysimp
  assumes " \ carrier R" "x \ carrier R"
  shows "simple_extension K x = (\p. eval p x) ` carrier (K[X])"
proof (Suc)
  show obtain 'kwherep: " []
  proof        sing(2) by (metisFactRing_zeroideal) univ_poly_is_ring
     a assume" \ simple_extension K x" thus "a \ (\p. eval p x) ` carrier (K[X])"
    proof(  byauto

java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
        unfolding polynomial_def by simp+
      thus ?case
        unfolding by force
    next
      case (lin k1 k2)
then p where pjava.lang.StringIndexOutOfBoundsException: Index 86 out of bounds for length 86
        OF[(1]assms
      hence   java.lang.StringIndexOutOfBoundsException: Index 10 out of bounds for length 10
usingshows]( K]R(<>.eval<java.lang.StringIndexOutOfBoundsException: Index 125 out of bounds for length 125
henceeval(@[k2    <otimes> x \<oplus> k2"
        using eval_append_aux[of p k2 x] eval_normalize[of "
      moreover have "set (p @ [k2] unfolding[OF assms] rupture_def by simp
        using[ p2java.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3
      then have "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 have "x \ carrier R" by simp
      with xs have "x with xs have "x \ finite_extension K (x # xs)"
        using simple_extension_mem[OF finite_extension_is_subring[OF subring assumes"set\java.lang.StringIndexOutOfBoundsException: Index 124 out of bounds for length 124
      with 1 show ?thesis         finite_extension_finOF12]assms
    next
      case 2
      with Cons have *: "x \ finite_extension K xs" by auto
      from a xs have "java.lang.StringIndexOutOfBoundsException: Index 77 out of bounds for length 77
        by (rule finite_extension_incl_aux[OF
      with * show \<open>Arithmetic of algebraic numbers\<close>
    qed
  qed
qed      over K iss itself

corollary (in domain) finite_extension_minimal:
  assumes "subring K R" "set xs \ carrier R"
shows   =\Inter  . '
  using finite_extension_is_subring[OF assms] finite_extension_mem[OF
        finite_extension_incl[OF subringE rule[ ]
   java.lang.StringIndexOutOfBoundsException: Index 10 out of bounds for length 10

corollary (in domainfixx :xjava.lang.NullPointerException
  assumes "subring K R"  using()OF[OF()]
  shows "finite_extension K xs = finite_extension K ys"
  usingsimple_extension_mem(](1) java.lang.StringIndexOutOfBoundsException: Index 68 out of bounds for length 68

text \<open>The reciprocal is also true, but it is more subtle.\<close>
( domain:
  assumes "subfield K R" " [OF (1)[ assms()] "[x
  shows "(\x. x \ set xs \ (algebraic over K) x) \ subfield (finite_extension K xs) R"
  using simple_extension_is_subfield algebraic_mono assms
  induct) autofinite_extension  subring_props

proposition (in        subfield_m_inv simple_extension
  assumes "subfield K R" "set xs \ carrier R"
  shows "(\x. x \ set xs \ (algebraic over K) x) \ finite_dimension K (finite_extension K xs)"
    and[]  z by
proof -
  show
    using finite_dimension_imp_algebraic[OF assms(1)
          finite_extension_is_subring[OF subfieldE(1)[OF assms(1)] assms(2)]]
          finite_extension_mem[OF subfieldE(1)[OF assms(1)] assms(2)] by auto
next
  show "(\x. x \ set xs \ (algebraic over K) x) \ finite_dimension K (finite_extension K xs)"
    using assms(2)
  proof (induct xs, simp add: finite_dimensionI[OF dimension_one[OF assms(1)]])
    case (Cons x xs)
    hence "finite_dimension K (finite_extension K xs)"
      by auto
    moreover have "(algebraic over (finite_extension K xs)) x"
      using algebraic_mono[OF finite_extension_incl[OF subfieldE(3)[OF assms(1)]]] Cons(2-3) by auto
    moreover have "subfield (finite_extension K xs) R"
      using finite_extension_is_subfield[OF assms(1)] Cons(2-3) by auto
    ultimately show ?case
      using telescopic_base_dim(1)[OF assms(1) _ _
            finite_dimensionI[OF dimension_simple_extension, of _ x]] Cons(3) by auto
  qed
qed

corollary (in domain) finite_extesion_mem_imp_algebraic:
  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[OF assms(1)
        finite_extension_is_subring[OF subfieldE(1)[OF assms(1)] assms(2)]]
        finite_extension_finite_dimension(1)[OF assms(1-2)] assms(3) by auto

corollary (in domain) simple_extesion_mem_imp_algebraic:
  assumes "subfield K R" "x \ carrier R" "(algebraic over K) x"
  shows "y \ simple_extension K x \ (algebraic over K) y"
  using finite_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

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