why $F(a^2) \subset F(a)$? I have some  doubt in this post
Let $E$ be an extension field of $F$. If $a \in E$ has a minimal polynomial of odd degree over $F$, show that $F(a)=F(a^2)$.
let $n$ be the degree of the minimal polynomial $p(x)$ of $a$ over $F$ and $k$ be the degree of the minimal polynomial $q(x)$ of $a^2$ over $F$
Since $a^2 \in F(a)$, We have $F(a^2) \subset F(a)$, then $k\le n$
I don't understand  why  $F(a^2) \subset F(a)$?
My thinking: Take $ a  \in \mathbb{R} $, $a \subset a^2 \implies F(a) \subset F(a^2)$
Credit : Atticus Stonestrom( comment section)
For  $a = \sqrt 2$, $ F(a)=\mathbb{Q}({\sqrt 2})$ contain basis  $\{1, \sqrt 2\}$ but  $ F(a^2)=\mathbb{Q}(2)$ contain basis $1 \implies F(a^2) \subset F(a)$
Credit : Atticus Stonestrom( comment section)
For  $a = \sqrt 2$, $ F(a)=\mathbb{Q}({\sqrt 2})$ contain basis  $\{1, \sqrt 2\}$ but  $ F(a^2)=\mathbb{Q}(2)$ contain basis $1 \implies F(a^2) \subset F(a)$