
theorem thX1ee:
for p being Prime
for n being non zero Nat
for F being p-characteristic Field st card F = p|^n
holds Roots X^(p|^n,F) = the carrier of F
proof
let p be Prime, n be non zero Nat, F be p-characteristic Field;
assume AS: card F = p|^n;
A: now let o be object;
   assume o in the carrier of F;
   then reconsider a = o as Element of F;
   a is_a_root_of X^(p|^n,F) by AS,thX1;
   hence o in Roots X^(p|^n,F) by POLYNOM5:def 10;
   end;
for o being object st o in Roots X^(p|^n,F) holds o in the carrier of F;
hence thesis by A,TARSKI:2;
end;
