Exercise 11, § 10

a) Let E be a separable extension of a field K of finite degree and N the Galois extension of K generated by E. If $a$ is an element of N whose conjugates form a normal basis of N over K and if $\beta = \mathrm{Tr}_{N/E}(a)$, show that $E = K(\beta)$. b) Let N be a Galois extension of K of finite degree and let $a$ be an element of N whose conjugates form a normal basis over K. If L is a subextension of N which is Galois over K and if $\beta = \mathrm{Tr}_{N/L}(a)$, show that the conjugates of $\beta$ form a normal basis of L over K. c) Let L, M be two Galois extensions of K of finite degree, such that $L \cap M = K$; let $\alpha$ (resp. $\beta$) be an element of L (resp. M) whose conjugates over K form a normal basis of L (resp. M) over K ; show that the conjugates over K of $\alpha \beta$ form a normal basis of $K(L \cup M)$.