Let $F$ and $K$ be fields. If $F\subseteq K$ and $r\in K$ s.t. $r^2$ is algebraic over $F$. Then $r$ is algebraic over $F$.

Question

Let $F$ and $K$ be fields. If $F\subseteq K$ and $r\in K$ s.t. $r^2$ is algebraic over $F$. Then $r$ is algebraic over $F$.

Assume polynomial $p(x)\in F[x]$ s.t. $p(r^2)=0$

If $r\in K$ and $r^2$ is algebraic over $F$ I'm looking for a polynomial $p(x)\in F[x]$ s.t. $p(r)=0$? Or, does it not have to be the same polynomial? How do I know if I'm to use the same polynomial for $r$ and $r^2$, or different ones?

Answer 1

$\left(\left(\sqrt[4]{2}\right)^2\right)^2-2=0\Rightarrow (\sqrt[4]{2})^4-2=0$

Answer 2

Let $\;r^2\;$ be a zero of $\;f(x)\in F[x]\;$ :

$$f(x)=\sum_{k=0}^n a_kx^k\implies 0=f(r^2)=\sum_{k=0}^na_k(r^2)^k=\sum_{k=0}^na_kr^{2k}$$