
theorem contr:
for p being Prime
for F being p-characteristic Field
for a being Element of F
st not a in F|^p holds X^(p,a) is irreducible inseparable
proof
let p be Prime, F be p-characteristic Field, a be Element of F;
assume A0: not a in F|^p;
the carrier of F|^p = the set of all a|^p where a is Element of F by deffp;
then not ex b being Element of F st b|^p = a by A0;
hence X^(p,a) is irreducible by pirred;
consider E being FieldExtension of F such that
A2: X^(p,a) is_with_roots_in E by FIELD_5:30;
consider b being Element of E such that
A3: b is_a_root_of X^(p,a),E by A2,FIELD_4:def 3;
F is Subfield of E by FIELD_4:7; then
the carrier of F c= the carrier of E by EC_PF_1:def 1; then
reconsider aE = a as Element of E;
A4: X^(p,a) = X^(p,aE) by split2;
0.E = Ext_eval(X^(p,a),b) by A3,FIELD_4:def 2
   .= eval(X^(p,aE),b) by split2,FIELD_4:26; then
b is_a_root_of X^(p,aE); then
b|^p = aE by split1; then
X^(p,aE) = (X-b)|^p by split0 .= (X-b)`^p; then
p = multiplicity(X^(p,aE),b) by ro0
 .= multiplicity(X^(p,a),b) by A4,FIELD_14:def 6;
hence thesis by INT_2:def 4,ThSep02;
end;
