Equal simple field extensions? I have a question about simple field extensions.
For a field $F$, if $[F(a):F]$ is odd, then why is $F(a)=F(a^2)$?

Note that $F(a)\supseteq F(a^2)\supseteq F$, hence
$$\text{odd }=[F(a):F]=[F(a):F(a^2)][F(a^2):F].$$
What can $[F(a):F(a^2)]$ be? What, therefore, must it be?
