Exercise 7, § 11

An extension E of a field K is called a Kummer extension if there exists an integer $n > 0$ such that K contains a primitive n-th root of unity and E is an abelian extension with Galois group annihilated by n. a) Let K be a field and N a Galois extension of degree m of K with Galois group $\Gamma$. For N to be a Kummer extension of K it is necessary and sufficient that there should exist a basis $(\theta_\sigma)_{\sigma \in \Gamma}$ of N over K such that the elements $\gamma_{\sigma \tau} = \theta_\sigma^{1-\tau}$ belong to K for each pair $(\sigma, \tau)$. (For the necessity of the condition reduce to the case where N is cyclic of prime power degree. For the sufficiency observe that if $\tau \in \Gamma$ is of order $n$, then there exists $\sigma \in \Gamma$ such that $\gamma_{\sigma \tau}$ is a primitive n-th root of unity.) b) If $(\theta_\sigma)_{\sigma \in \Gamma}$ is a basis satisfying condition a), show that for an element $x = \sum \sigma a_\sigma \theta_\sigma$ of N to be such that the conjugates of x form a normal basis of N over K, it is necessary and sufficient that $a_\sigma \neq 0$ for all $\sigma \in \Gamma$.