
theorem ThBas:
for F being Field,
    E being FieldExtension of F
for a being Element of E st not a in F & a^2 in F
holds {1.E,a} is Basis of VecSp(FAdj(F,{a}),F) & deg(FAdj(F,{a}),F) = 2
proof
let F be Field, E be FieldExtension of F;
let a be Element of E;
assume AS: not a in F & a^2 in F;
reconsider a1 = a as F-algebraic Element of E by AS,lemBas0;
reconsider b = a^2 as Element of F by AS;
B: deg MinPoly(a1,F) = deg(X^2-b) by AS,lemBas1 .= 2 by FIELD_9:def 4;
Base a1 = {1.E,a}
   proof
   C: now let o be object;
      assume o in Base a1; then
      consider n being Element of NAT such that
      D1: o = a1|^n & n < 2 by B;
      per cases by D1,NAT_1:23;
      suppose n = 0;
        then o = 1_E by D1,BINOM:8;
        hence o in {1.E,a} by TARSKI:def 2;
        end;
      suppose n = 1;
        then o = a by D1,BINOM:8;
        hence o in {1.E,a} by TARSKI:def 2;
        end;
      end;
   now let o be object;
     assume o in {1.E,a}; then
     per cases by TARSKI:def 2;
     suppose o = 1.E;
       then o = 1_E .= a1|^0 by BINOM:8;
       hence o in Base a1 by B;
       end;
     suppose o = a;
       then o = a1|^1 by BINOM:8;
       hence o in Base a1 by B;
       end;
     end;
   hence thesis by C,TARSKI:2;
   end;
hence {1.E,a} is Basis of VecSp(FAdj(F,{a}),F) by FIELD_6:65;
thus thesis by B,FIELD_6:67;
end;
