Exercise 1, § 11

Let K be a field of characteristic p, n an integer not a multiple of p, $R_n(K)$ the field of n-th roots of unity and E a cyclic extension of $R_n(K)$ of degree n over K ; E is thus the root field of a polynomial $X^n - a \in R_n(K)[X]$ (V, p. 89, Th. 4). For E to be a Galois extension of K it is necessary and sufficient that for every automorphism $\tau \in \mathrm{Gal}(R_n(K)/K)$ there exists an integer $r > 0$ and an element $\gamma \in R_n(K)$ such that $\tau(\alpha) = \gamma^n \alpha'$.