Exercise 4, § 11

Let K be a field of characteristic p, q a prime number $\neq p$ such that K contains the q-th roots of unity and let $\zeta$ be a primitive q-th root of unity. a) Let E be a cyclic extension of K of degree $q^e$ and $\sigma$ a K-automorphism of E generating $\mathrm{Gal}(E/K)$. Let F be the field of degree $q^{e-1}$ over K such that $K \subset F \subset E$; there exists $0 \in E$, a root of an irreducible polynomial $X^q - a$ in $F[X]$ such that $E = F(0)$ and $\theta^{\sigma^m} = \zeta \theta$ (where $m = q^{e-1}$). Show that also $E = K(0)$ and $0'' = \beta \theta$ where $\beta \in F$ is such that $\alpha^{\sigma^{-1}} = \beta^q$ and $N_{F/K}(\beta) = \zeta$ (observe that $0'' = \beta \theta^k$ with $0 < k < q$ and $\beta \in F$, by Ex. 1 ; by calculating $\theta^{\sigma^{mq}}$, show that $k^{mq} - 1 \equiv 0 \pmod{q}$ and deduce that $k = 1$). b) Conversely, let F be a cyclic extension of degree $q^{e-1}$ of K with $e \geq 2$ and let $a$ be a K-automorphism of F generating $\mathrm{Gal}(F/K)$. If there exists an element $\beta \in F$ such that $N_{F/K}(\beta) = \zeta$ and if $\alpha \in F$ is such that $\alpha^{\sigma^{-1}} = \beta^q$, show that for every $c \in K^*$ the polynomial $X^q - c \alpha$ is irreducible in $F[X]$. If 8 is one of the roots of this polynomial in an algebraic closure $\Omega$ of F, show that there exists a K-homomorphism $\overline{\sigma}$ of $E = F(0)$ into $\Omega$, extending a and such that $8'' = \beta \theta$; deduce that E is a cyclic extension of K of degree $q^e$, that $\overline{\sigma}$ generates $\mathrm{Gal}(E/K)$ and that $E = K(0)$. Finally show that every cyclic extension of degree $q^e$ of K, containing F, is the root field of a polynomial $X^q - c \alpha$ in $F[X]$, for an appropriate $c \in K^*$ (use Th. 3 of V, p. 85). c) Take K to be the field Q of rational numbers. The polynomial $X^2 + 1$ is irreducible in $Q[X]$; if i is one of its roots, then $F = Q(i)$ is a cyclic extension of Q of degree 2 but there exists no cyclic extension of degree 4 of Q containing F.