
theorem
for p being Prime
for n being non zero Nat
for F being GaloisField of p|^n holds deg(F,Z/p) = n
proof
let p be Prime, n be non zero Nat, F be GaloisField of p|^n;
set V = VecSp(F,Z/p);
[#]V = the carrier of F by FIELD_4:def 6; then
C: V is finite-dimensional by RANKNULL:4;
consider T being linear-transformation of V,(Z/p)^*(dim V) such that
E: T is bijective by C,VECTSP13:30,VECTSP13:def 10;
F: dom T = the carrier of V by FUNCT_2:def 1;
G: rng T = the carrier of (Z/p)^*(dim V) by E,FUNCT_2:def 3;
card dom T = card rng T by E,CARD_1:70; then
A: card(the carrier of V)
     = card(Z/p) |^ (dim V) by F,G,VECTSP13:31
    .= p |^ (dim V) by fresh3a;
card the carrier of V = order F by FIELD_4:def 6 .= p|^n by defGal;
then dim V = n by A,lemp;
hence thesis by C,FIELD_4:def 7;
end;
