Exercise 15, § 10

Let E be a Galois extension of finite degree $n$ of a field K whose Galois group G is nilpotent. Let P be the set of prime numbers dividing n. Then there exist Galois subextensions $E_p$ of E ($p \in P$) such that $\mathrm{Gal}(E_p/K)$ is a p-group for each $p \in P$, and such that the canonical homomorphism $\bigotimes_{p \in P} E_p \to E$ is an isomorphism.