
theorem FA4a:
for F being finite Field
for f being Automorphism of F
for a being Element of F
holds f.a in PrimeField F iff a in PrimeField F
proof
let F be finite Field, f be Automorphism of F, a be Element of F;
consider n being non zero Nat such that
AS: card F = (Char F)|^n by FIELD_15:92;
set p = Char F, P = PrimeField F;
reconsider F as p-characteristic Field by RING_3:def 6;
id F is isomorphism; then
H: F is F-homomorphic Ring by RING_2:def 4;
A: now assume f.a in P; then
   f.a = (f.a)|^p by AS,fresh3P .= f.(a|^p) by H,lemID; then
   a = a|^p by FUNCT_2:19;
   hence a in PrimeField F by AS,fresh3P;
   end;
now assume a in P;
  then a = a|^p by AS,fresh3P;
  then f.a = (f.a)|^p by H,lemID;
  hence f.a in P by AS,fresh3P;
  end;
hence thesis by A;
end;
