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 $F(a)=F(a^2)$?

Firstly, $F(a^2)\subseteq F(a)$. If the inclusion is strict, then $[F(a):F(a^2)]\neq 1$. Now $a$ satisfies the polynomial $X^2-a^2\in F(a^2)[X]$, so the minimal polynomial $m_{a,F(a^2)}(x)$ has degree at most $2$, and this is necessarily $2$, so $[F(a):F(a^2)]=2$. 
Then since the degree of field extensions is multiplicative, you get $2\mid [F(a):F]$, a contradiction.
