
theorem
for F being Field holds F is finite iff
ex p being Prime, n being non zero Nat,
   G being GaloisField of p|^n st F,G are_isomorphic
proof
let F be Field;
A: now assume F is finite; then
   reconsider F1 = F as finite Field;
   consider p being Prime, n being non zero Nat such that
   A: Char F1 = p & order F1 = p|^n by finex2;
   set G = the GaloisField of p|^n;
   order G = p|^n by defGal;
   hence ex p being Prime, n being non zero Nat,
            G being GaloisField of p|^n st F,G are_isomorphic
     by A,finun;
  end;
now given p being Prime, n being non zero Nat,
          G being GaloisField of p|^n such that
  A: F,G are_isomorphic;
  consider h being Function of F,G such that
  B: h is isomorphism by A;
  dom h = the carrier of F & rng h = the carrier of G
     by B,FUNCT_2:def 3,FUNCT_2:def 1; then
  card the carrier of F = card the carrier of G by B,CARD_1:70;
  hence F is finite;
  end;
hence thesis by A;
end;
