
theorem sp:
for F being Field
for a being Element of F
for n being non zero Nat holds (X-a)`^n splits_in F
proof
let F be Field, a be Element of F, n be non zero Nat;
defpred P[Nat] means (X-a)`^($1) splits_in F;
H1: deg(X-a) = 1 & (X-a)`^1 = (X-a) by FIELD_5:def 1,POLYNOM5:16; then
IA: P[1] by FIELD_4:29;
IS: now let k be Nat;
    assume k >= 1;
    assume P[k]; then
    H2: (X-a)`^k splits_in F & (X-a) splits_in F by H1,FIELD_4:29;
    (X-a)`^(k+1) = ((X-a)`^k) *' (X-a) by POLYNOM5:19;
    hence P[k+1] by H2,FIELD_8:12;
    end;
I: for k being Nat st k >= 1 holds P[k] from NAT_1:sch 8(IA,IS);
n >= 0 + 1 by INT_1:7;
hence thesis by I;
end;
