Exercise 8, § 10

Let K be a field and N a Galois extension of K of finite degree $n$. a) Let $p$ be a prime number dividing $n$ and let $p'$ be the highest power of $p$ dividing $n$. Show that there exist subextensions $L_i$ of N for $0 \leq i \leq r$ such that for $0 \leq i \leq r - 1$, $L_i$ contains $L_{i-1}$, $L_0 = N$, $L_i$ is a Galois extension of degree $p$ of $L_{i-1}$, and $[L_i : K]$ is not divisible by $p$ (cf. I, p. 77, Th. 1 and I, p. 78, Th. 2). b) Let $x \in N$ be such that N is generated by x and its conjugates. Show that there exists a subextension E of N such that $x \notin E$ and $[N : E] = p'$. (Consider the Sylow p-groups $H_j$ of $\mathrm{Gal}(N/K)$ and the (normal) subgroup H generated by their union ; if $F_j$ (resp. F) is the subfield of invariants of H, (resp. H) show that $x \notin F$ and deduce that there is an index j such that $x \notin F_j$; then we can take $E = F_j$.) Conclude that there exists a subextension L of N such that $L(x)$ is a Galois extension of degree $p$ of L (use I, p. 77, Prop. 12). c) Conversely, let $\Omega$ be an algebraic closure of N and suppose that there exists a subextension L of $\Omega$ such that $L(x)$ is Galois of degree $p$ over L. Show that then $p$ divides $n$. (Observe that $N \cap L(x) = (N \cap L)(x)$ and that $(N \cap L)(x)$ is a Galois extension of degree $p$ of $N \cap L$.)