
theorem
for F being Field,
    E being (Polynom-Ring F)-homomorphic FieldExtension of F
for a being non zero Element of E holds a is F-algebraic iff a" in RAdj(F,{a})
proof
let F be Field, E be (Polynom-Ring F)-homomorphic FieldExtension of F;
let a be non zero Element of E;
X: F is Subring of E by FIELD_4:def 1;
A: now assume a is F-algebraic;
   then A1: FAdj(F,{a}) = RAdj(F,{a}) by ch1;
   A2: {a} is Subset of FAdj(F,{a}) by FAt;
   a in {a} by TARSKI:def 1; then
   reconsider b=a as Element of the carrier of FAdj(F,{a}) by A2;
   b" in the carrier of FAdj(F,{a});
   hence a" in RAdj(F,{a}) by A1,Th19f;
   end;
now assume a" in RAdj(F,{a}); then
   a" in the set of all Ext_eval(p,a) where p is Polynomial of F by lemphi5;
   then consider p being Polynomial of F such that
   A1: a" = Ext_eval(p,a);
   set r = (-1.F)|F;
   set q = rpoly(1,0.F) *' p + r;
   -0.F = 0.F; then
   -1.F <> 0.F; then
   B0: deg r = 0 by RING_4:21;
   B5: deg rpoly(1,0.F) = 1 by HURWITZ:27;
   q is non zero
     proof
     per cases;
     suppose p = 0_.(F);
       hence thesis by B0,HURWITZ:20;
       end;
     suppose B3: p <> 0_.(F);
       then reconsider degp = deg p as Element of NAT by FIELD_1:1;
       B4: deg(rpoly(1,0.F) *' p)
              = deg rpoly(1,0.F) + degp by B3,HURWITZ:23;
       deg q = max(deg rpoly(1,0.F)+deg p,0) by B0,B4,B5,HURWITZ:21
            .= deg rpoly(1,0.F) + degp by XXREAL_0:def 10;
       then q <> 0_.(F) by HURWITZ:20;
       hence thesis by UPROOTS:def 5;
       end;
     end;
   then reconsider q as non zero Polynomial of F;
   1.E = 1.F by X,C0SP1:def 3; then
   A2: -1.E = (-1.F) * 1.F by X,Th19
           .= (-1.F)*LC(1_.(F)) by RATFUNC1:def 7
           .= LC((-1.F)*1_.(F)) by RATFUNC1:18
           .= LC((-1.F)|F) by RING_4:16;
   A3: 0.E = 0.F by X,C0SP1:def 3;
   reconsider rpE = rpoly(1,0.E) as
        Element of the carrier of Polynom-Ring E by POLYNOM3:def 10;
   reconsider rpF = rpoly(1,0.F) as
        Element of the carrier of Polynom-Ring F by POLYNOM3:def 10;
   A5: a <> 0.E;
   Ext_eval(q,a) = Ext_eval(rpoly(1,0.F) *' p,a) + Ext_eval(r,a)
                   by X,ALGNUM_1:15
                .= Ext_eval(rpoly(1,0.F),a) * a" + Ext_eval(r,a)
                   by X,A1,ALGNUM_1:20
                .= Ext_eval(rpF,a) * a" + -1.E by A2,exevalconst
                .= eval(rpE,a) * a" + - 1.E by A3,FIELD_4:21,FIELD_4:26
                .= (a - 0.E) * a" + - 1.E by HURWITZ:29
                .= a" * a + - 1.E by GROUP_1:def 12
                .= 1.E + - 1.E by A5,VECTSP_1:def 10
                .= 0.E by RLVECT_1:5;
   hence a is F-algebraic by alg00;
   end;
hence thesis by A;
end;
