Let F be a field, and suppose that alpha is algebraic over F. Prove that if [F(alpha):F] is odd, then F(a^2)=F(a).

Hows this look: We know that F(a^2)=F(a) if and only if [F(a):F(a^2)]=1. If [F(a):F(a^2)] does not equal 1. Then [F(a):F(a^2)]=2. By the tower law this contradicts that [F(alpha):F] is odd.

Any corrections and things to add would be great! Thanks.