
theorem FA3:
for p being Prime
for n being non zero Nat
for F being Field
for E being FieldExtension of F
st card E = p|^n & F == PrimeField E holds deg(E,F) = n
proof
let p be Prime, n be non zero Nat, F be Field;
let E be FieldExtension of F;
assume AS: card E = p|^n & F == PrimeField E; then
A: E is finite & F is Subfield of E by FIELD_4:7; then
reconsider F1 = F as finite Field;
reconsider E1 = E as FieldExtension of F1;
reconsider P = PrimeField E as finite Field by A;
set V = VecSp(E1,F1);
[#]V = the carrier of E1 by FIELD_4:def 6; then
[#]V is finite by AS; then
C: V is finite-dimensional by RANKNULL:4;
consider T being linear-transformation of V,F1^*(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 F1^*(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(P) |^ (dim V) by AS,F,G,VECTSP13:31
    .= p |^ (dim V) by AS,FA2;
card(the carrier of V) = p|^n by AS,FIELD_4:def 6; then
dim V = n by A,lemp;
hence thesis by C,FIELD_4:def 7;
end;
