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Problem: Prove that every algebraic extension of a finite field is normal.

Attempt: Let $\mathbb{F}_q $ be a finite field, with $q = p^n$, where $p$ is prime, and let $\mathbb{F}_q \subset E$ be a (finite) algebraic extension.

My attempt was to show somehow that $E = \mathbb{F}_{q^m} $ for some $m \in \mathbb{N}$. Then I also know that $E =\mathbb{F}_{q^m}$ is the splitting field of $$x^{q^{m}} - x = \prod_{\beta \in \mathbb{F}_{q^m}} (x - \beta). $$ This would show that the extension is normal. But I'm not sure how to show that $E = \mathbb{F}_{q^m}$.

Help is appreciated!

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  • $\begingroup$ Hint. Remember that $E$ is a (finite-dimensional) vector space over $\mathbb{F}_q$. $\endgroup$ – Sofie Verbeek Aug 30 '18 at 17:13
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Hint: $E$ is an $F_q$ vector space of dimension $m$. This impies that $|E|=|F_q|^m$. To sse this take a basis of the $F_q$-vector space $E$ and counts the elements of $E$.

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  • $\begingroup$ Beat me by four seconds! $\endgroup$ – Sofie Verbeek Aug 30 '18 at 17:13

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