## 9.18 Finite fields

Let $F$ be a finite field. It is clear that $F$ has positive characteristic as we cannot have an injection $\mathbf{Q} \to F$. Say the characteristic of $F$ is $p$. The extension $\mathbf{F}_ p \subset F$ is finite. Hence we see that $F$ has $q = p^ f$ elements for some $f \geq 1$.

Let us think about the group of units $F^*$. This is a finite abelian group, so it has some exponent $e$. Then $F^* = \mu _ e(F)$ and we see from the discussion in Section 9.17 that $F^*$ is a cyclic group of order $q - 1$. (A posteriori it follows that $e = q - 1$ as well.) In particular, if $\alpha \in F^*$ is a generator then it clearly is true that

$F = \mathbf{F}_ p(\alpha )$

In other words, the extension $F/\mathbf{F}_ p$ is generated by a single element. Of course, the same thing is true for any extension of finite fields $E/F$ (because $E$ is already generated by a single element over the prime field).

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