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Lemma 9.24.2. Let $K$ be a field with algebraic closure $\overline{K}$. Let $p$ be a prime different from the characteristic of $K$. Let $\zeta \in \overline{K}$ be a primitive $p$th root of $1$. Then $K(\zeta )/K$ is a Galois extension of degree dividing $p - 1$.

Proof. The polynomial $x^ p - 1$ splits completely over $K(\zeta )$ as its roots are $1, \zeta , \zeta ^2, \ldots , \zeta ^{p - 1}$. Hence $K(\zeta )/K$ is a splitting field and hence normal. The extension is separable as $x^ p - 1$ is a separable polynomial. Thus the extension is Galois. Any automorphism of $K(\zeta )$ over $K$ sends $\zeta $ to $\zeta ^ i$ for some $1 \leq i \leq p - 1$. Thus the Galois group is a subgroup of $(\mathbf{Z}/p\mathbf{Z})^*$. $\square$

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