Equal simple field extensions?
Solution 1:
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.
Solution 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?