
theorem
for F being Field
for E being FieldExtension of F
for a being F-algebraic Element of E
holds card(the set of all f where f is F-fixing Automorphism of FAdj(F,{a}))
      c= card Roots(FAdj(F,{a}),MinPoly(a,F))
proof
let F be Field, E be FieldExtension of F, a be F-algebraic Element of E;
set M = the set of all f where f is F-fixing Automorphism of FAdj(F,{a});
set R = Roots(FAdj(F,{a}),MinPoly(a,F));
a in {a} & {a} is Subset of FAdj(F,{a}) by TARSKI:def 1,FIELD_6:35; then
reconsider a1 = a as Element of FAdj(F,{a});
   D: E is FAdj(F,{a})-extending by FIELD_4:7;
   0.FAdj(F,{a}) = 0.E by FIELD_6:def 6
    .= Ext_eval(MinPoly(a,F),a) by FIELD_6:52
    .= Ext_eval(MinPoly(a,F),a1) by D,FIELD_6:11; then
   a1 is_a_root_of MinPoly(a,F),FAdj(F,{a}) by FIELD_4:def 2; then
   a1 in {b where b is Element of FAdj(F,{a}) :
                      b is_a_root_of MinPoly(a,F),FAdj(F,{a})}; then
H: Roots(FAdj(F,{a}),MinPoly(a,F)) <> {} by FIELD_4:def 4;
defpred P[object,object] means
  ex g being F-fixing Automorphism of FAdj(F,{a}) st $1 = g & $2 = g.a;
B: now let x be object;
   assume x in M; then
   consider g being F-fixing Automorphism of FAdj(F,{a}) such that
   B1: x = g;
   g.a in R by ID2a;
   hence ex y being object st y in R & P[x,y] by B1;
   end;
consider h being Function of M,R such that
A: for o being object st o in M holds P[o,h.o] from FUNCT_2:sch 1(B);
   now let x1,x2 be object;
   assume A0: x1 in M & x2 in M & h.x1 = h.x2; then
   consider g1 being F-fixing Automorphism of FAdj(F,{a}) such that
   A1: x1 = g1 & h.x1 = g1.a by A;
   consider g2 being F-fixing Automorphism of FAdj(F,{a}) such that
   A2: x2 = g2 & h.x2 = g2.a by A,A0;
   thus x1 = x2 by A0,A1,A2,ID;
   end; then
B: h is one-to-one by H,FUNCT_2:19;
C: rng h c= R by RELAT_1:def 19;
dom h = M by H,FUNCT_2:def 1;
hence thesis by B,C,CARD_1:10;
end;
