Exercise 9, § 11

Let K be a field of characteristic $p > 0$. a) Let E be a cyclic extension of degree $p^e$ of K and let $\sigma$ be a K-automorphism generating the group Gal(E/K). Let F be the field of degree $p^{e-1}$ over K such that $K \subset F \subset E$; if $m = p^{e-1}$, show that there exists $0 \in E$, root of an irreducible polynomial $X^p - X - \alpha$ of F[X] such that $E = F(\theta)$ and $\sigma^m(\theta) = 0 + 1$. Show that we also have $E = K(\theta)$ and $\sigma(\theta) = 0 + \beta$ where $\beta \in F$ is such that $\sigma(\alpha) - \alpha = \beta^p - \beta$ and $\operatorname{Tr}_{F/K}(\beta) = 1$. b) Conversely, let F be a cyclic extension of degree $p^{e-1}$ of K (with $e > 1$), $\sigma$ a K-automorphism of F generating Gal(F/K). Show that there exist two elements $a, \beta$ of F such that $\operatorname{Tr}_{F/K}(\beta) = 1$ and $\sigma(\alpha) - a = \beta^p - \beta$ (use Th. 3 of V, p. 85 and the Cor. of Prop. 1 of V, p. 49). Deduce that for all $c \in K$ the polynomial $X^p - X - a - c$ is irreducible in F[X]; if 0 is a root of this polynomial in an algebraic closure $\Omega$ of F, show that there exists a K-homomorphism $a$ of $E = F(\theta)$ in $\Omega$ extending $\sigma$ and such that $\bar{\sigma}(\theta) = 0 + \beta$; conclude that E is a cyclic extension of degree $p^e$ of K, that a generates Gal(E/K) and that $E = K(0)$. Finally show that every cyclic extension of degree $p^e$ of K containing F is the splitting field of a polynomial $X^p - X - \alpha - c$ of F[X] for a suitable $c \in K$.