
theorem
for p being Prime
for n being non zero Nat
for F being finite Field st card F = p|^n
holds X^(p|^n,F) is Ppoly of F,[#](the carrier of F)
proof
let p be Prime, n be non zero Nat, F be finite Field;
assume AS: card F = p|^n;
consider n1 being non zero Nat such that
A: card F = (Char F)|^n1 by FIELD_15:92;
Char F = p by A,AS,lemp; then
H: F is p-characteristic by RING_3:def 6;
set q = X^(p|^n,F);
F is SplittingField of X^(p|^n,PrimeField F) by AS,split; then
X^(p|^n,PrimeField F) splits_in F by FIELD_8:def 1; then
consider x being non zero Element of F, v being Ppoly of F such that
I: X^(p|^n,PrimeField F) = x * v by FIELD_4:def 5;
F is FieldExtension of (PrimeField F) by FIELD_4:7; then
E: q = X^(p|^n,PrimeField F) by Xm; then
consider a being non zero Element of F, q1 being Ppoly of F such that
A: q = a * q1 by I;
B: a = a * 1.F
    .= a * (LC q1) by RING_5:50
    .= LC q by A,RING_5:5
    .= 1.F by Lm13;
C: now let a be Element of F;
   assume D1: a is_a_root_of q;
   reconsider F1 = F as FieldExtension of F by FIELD_4:6;
   reconsider a1 = a as Element of F1;
   0.F = Ext_eval(q,a1) by D1,FIELD_4:26; then
   D2: a1 is_a_root_of q,F1 by FIELD_4:def 2;
   thus multiplicity(q,a) = multiplicity(q,a1) by FIELD_14:def 6
                        .= 1 by D2,H,E,I,FIELD_4:def 5,FIELD_15:75;
   end;
Roots q1 = the carrier of F by AS,H,A,B,thX1ee
        .= [#](the carrier of F) by SUBSET_1:def 3;
hence thesis by A,B,C,FIELD_14:30;
end;
