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We have a field extension $K/k$ and we're asked to prove that $\alpha \in K$ is algebraic over $k$ $\iff$ $\alpha^2$ is algebraic over $k$.

I've got an argument for the left arrow $\implies$, but I'm not sure how to proceed with $\impliedby$. Do we need to manipulate a certain polynomial or is it a linear algebra argument, perhaps?

Any help would be appreciated!

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If $\alpha^2$ is a root of $P(x)$ then $\alpha$ is root of $P(x^2)$.

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We have $\alpha^2 \in k[\alpha]$ where $k[\alpha]$ is the field where $\alpha$ is adjoined to $k$. Hence $k[\alpha^2] \subset k[\alpha]$ and we must have

$$[\space k[\alpha^2]:k] \leq [k[\alpha]:k \space]. $$

Therefore, if the extension $ k[\alpha]/k $ has finite degree, also the extension $k[\alpha^2]/k$ has finite degree. And finite degree means algebraic.

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