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