习题 8, § 10

设K为有限次数$n$的K域和N伽罗瓦扩张。 a) 令$p$为质数数除以$n$，并令$p'$为$p$除以$n$的最高幂。 证明对于 $0 \leq i \leq r$ 存在 N 的子扩展 $L_i$，使得对于 $0 \leq i \leq r - 1$、$L_i$ 包含 $L_{i-1}$、$L_0 = N$、$L_i$ 是次数 $p$ 的伽罗瓦扩张 $L_{i-1}$、$[L_i : K]$ 不能被 $p$整除（cf. I、p. 77、Th. 1 和 I、p. 78、Th. 2）。 b) 令 $x \in N$ 使得 N 由 x 及其共轭生成。 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. 证明然后 $p$ 除 $n$。 （观察 $N \cap L(x) = (N \cap L)(x)$ 且 $(N \cap L)(x)$ 是次数 $p$ 伽罗瓦扩张 $N \cap L$。）