
theorem mi1:
for F being Field
for p being Element of the carrier of Polynom-Ring F
holds (ex E being FieldExtension of F,
          a being F-algebraic Element of E st p = MinPoly(a,F)) iff
      (p is monic irreducible)
proof
let F be Field, p1 be Element of the carrier of Polynom-Ring F;
now assume p1 is irreducible monic;
then reconsider p = p1 as irreducible monic
                                 Element of the carrier of Polynom-Ring F;
set KF = KroneckerField(F,p), u = KrRoot p;
consider FP being Field such that
X1: KF,FP are_isomorphic &
    (the carrier of FP) /\ ((the carrier of KF) \/ (the carrier of F)) = {}
    by FIELD_5:29;
X: [#] F = the carrier of F & [#] FP = the carrier of FP;
X2: F,FP are_disjoint
    proof
    now assume A: (the carrier of F) /\ (the carrier of FP) <> {};
      set x = the Element of (the carrier of F) /\ (the carrier of FP);
      B: x in the carrier of F & x in the carrier of FP by A,XBOOLE_0:def 4;
      then x in (the carrier of KF) \/ (the carrier of F) by XBOOLE_0:def 3;
      hence contradiction by B,X1,XBOOLE_0:def 4;
      end;
    hence thesis by X,FIELD_2:def 1;
    end;
consider phi being Function of KF,FP such that
X3: phi is isomorphism by X1;
reconsider KroneckerIsop = (KroneckerIso p)" as
    Isomorphism of KroneckerField(F,p),Polynom-Ring p by RING_3:73;
set h = phi * (emb p);
reconsider h as Function of F,FP;
X4: h is linear one-to-one by X3,RINGCAT1:1; then
reconsider FP as F-monomorphic Field by RING_3:def 3;
reconsider h as Monomorphism of F,FP by X4;
reconsider E = embField h as Field by X2,FIELD_2:12;
emb_iso h is onto by X2,FIELD_2:15; then
reconsider embisoh = (emb_iso h)" as Function of FP,E by FUNCT_2:25;
emb_iso h is additive multiplicative unity-preserving
   by X2,FIELD_2:13,FIELD_2:14; then
Y: emb_iso h is linear one-to-one onto by X2,FIELD_2:15; then
reconsider FP as E-isomorphic Field by RING_3:def 4;
reconsider embisoh as Isomorphism of FP,E by Y,RING_3:73;
set iso = embisoh * phi;
reconsider iso as Function of KF,E;
X5: iso is RingHomomorphism by X3,RINGCAT1:1;
then reconsider E as KF-homomorphic Field by RING_2:def 4;
reconsider iso as Homomorphism of KF,E by X5;
u is_a_root_of emb(p,p) by FIELD_1:42; then
Z: eval((PolyHom iso).(emb(p,p)),iso.u) = 0.E by FIELD_1:33,POLYNOM5:def 7;
F is Subfield of E by FIELD_2:17;
then reconsider E as FieldExtension of F by FIELD_4:7;
take E;
reconsider a = iso.u as Element of E;
(PolyHom iso).(emb(p,p)) = p
   proof
   set g = (PolyHom iso).(emb(p,p));
   A: for a being Element of F holds iso.((emb p).a) = a
      proof
      let a be Element of F;
      reconsider b = a|F as Element of the carrier of Polynom-Ring F
          by POLYNOM3:def 10;
      reconsider v = Class(EqRel(Polynom-Ring F,{p}-Ideal),b)
          as Element of KroneckerField(F,p) by RING_1:12;
      dom(emb p) = the carrier of F by FUNCT_2:def 1; then
      C: h.a = phi.((emb p).a) by FUNCT_1:13 .= phi.v by FIELD_1:39;
      the carrier of (embField h) = carr h by FIELD_2:def 7
       .= ([#]FP \ rng h) \/ [#]F by FIELD_2:def 2
       .= ((the carrier of FP)\(rng h))\/(the carrier of F);
      then reconsider a1 = a as Element of embField h by XBOOLE_0:def 3;
      a in F; then
      D: (emb_iso h).a1 = phi.v by C,FIELD_2:def 8;
      A1: dom phi = the carrier of KF by FUNCT_2:def 1;
      A3: dom(emb_iso h) = the carrier of E by FUNCT_2:def 1;
      thus iso.((emb p).a) = (embisoh * phi).v by FIELD_1:39
        .= ((emb_iso h)").(phi.v) by A1,FUNCT_1:13
        .= a by D,A3,FUNCT_1:34;
      end;
   now let x be object;
     assume x in NAT;
     then reconsider i = x as Element of NAT;
     g.i = iso.(emb(p,p).i) by FIELD_1:def 2
        .= iso.(Class(EqRel(Polynom-Ring F,{p}-Ideal),(p.i)|F))
           by FIELD_1:40
        .= iso.((emb p).(p.i)) by FIELD_1:39;
     hence g.x = p.x by A;
     end;
   hence thesis;
   end;
then K: Ext_eval(p,a) = 0.E by Z,FIELD_4:26;
then reconsider a1 = a as F-algebraic Element of E by FIELD_6:43;
take a1;
p = MinPoly(a1,F) by K,FIELD_6:52;
hence thesis;
end;
hence thesis;
end;
