Exercise 8, § 11

Let N be a Kummer extension of a field K. a) Let E be a subextension of N such that [E : K] = q is a prime number. Show that there exists a decomposition $N = N_1 \otimes N_2 \otimes \ldots \otimes N_r$ of N into cyclic extensions of K such that $E \subset N_j$ for an index j. b) Deduce from a) that if $x \in N$ is such that the conjugates of x over K form a normal basis of N over K, then for each field E such that $K \subset E \subset N$, the conjugates of x over E form a normal basis of N over E (reduce to the case where [E : K] is prime and use Ex. 7, b)). \*