
theorem GP:
for p being Prime
for F being GaloisField of p holds F == Z/p
proof
let p be Prime, F be GaloisField of p;
A: order F = p|^1 & Z/p is Subfield of F by defGal; then
H: card F = card Z/p by fresh3a;
reconsider X = the carrier of F,
           Y = the carrier of Z/p as finite set;
B: Y c= X by A,EC_PF_1:def 1; then
card(X \ Y) = card X - card Y by CARD_2:44 .= 0 by H; then
X \ Y = {}; then
the carrier of F c= the carrier of Z/p by XBOOLE_0:def 5; then
E: the carrier of F = the carrier of Z/p by B,XBOOLE_0:def 10;
   the addF of Z/p = (the addF of F) || the carrier of Z/p &
   the multF of Z/p = (the multF of F) || the carrier of Z/p &
   1.F = 1.Z/p & 0.F = 0.Z/p by A,EC_PF_1:def 1;
hence thesis by E;
end;
