
theorem lemBas1:
for F being Field,
    E being FieldExtension of F
for a being F-algebraic Element of E st not a in F
for b being Element of F st b = a^2 holds MinPoly(a,F) = X^2-b
proof
let F be Field, E be FieldExtension of F,a be F-algebraic Element of E;
assume AS1: not a in F;
let b be Element of F;
assume AS2: b = a^2;
set p = X^2-b;
B: p is monic & deg p = 2 by FIELD_9:def 4;
C: Ext_eval(X^2-b,a) = 0.E by AS2,lemBas00;
for q being non zero Polynomial of F
  st Ext_eval(q,a) = 0.E holds deg p <= deg q by B,AS1,lemBas01;
hence thesis by C,FIELD_6:51;
end;
