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I don't understand a step in the proof of corollary 96 in the book "Galois Theory" by J. Rotman. The full book can be found here https://epdf.pub/galois-theory-second-edition-universitext.html

Corollary 96: Let $E / F$ be a Galois extension of prime degree $p$. If $F$ has a primitive $p$th root of unity, then $E = F(\beta)$, where $\beta^p \in F$, and so $E / F$ is a pure extension.

Here is the beginning of the proof:

Proof: If $\omega$ is a primitive $p$th root of unity, then $N(\omega) = \omega^p = 1$, because $\omega \in F$. Now $Gal(E/F) \simeq \mathbb{Z}_p$, by Corollary 71, hence is cyclic; let $\sigma$ be a generator. ...

($N$ denotes the norm.) And Corollary 71 says:

Corollary 71: Let $p$ be a prime, let $F$ be a field containing a primitive $p$th root of unity, and let $f(x) = x^p - c \in F[x]$ have splitting field $E$. Then either $f(x)$ splits and $Gal(E/F) = 1$ or it is irreducible and $Gal(E/F) \simeq \mathbb{Z}_p$.

My question: I do not see how $Gal(E/F) \simeq \mathbb{Z}_p$ follows from Corollary 71. Isn't it true that every group of prime order $p$ is cyclic and isomorph to $\mathbb{Z}_p$? Have I overseen something?

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You are correct, Corollary 71 is not relevant and you only have to observe that every group of order $p$ is cyclic.

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