Exercise 11, § 16

Let L be a cyclic extension of K with Galois group G, and let $\sigma$ be a generator of G. Exercise 10 gives a canonical isomorphism Br(L$/K$)$\rightarrow K^*/N_{L/K}(L^*)$, whose inverse we will now construct explicitly.

Let $\theta \in K^*$, and let $c_{\theta}\in Z^2(G,L^*)$ be the element defined by $c_{\theta}(\sigma^p, \sigma^q) =\theta^{\varepsilon(p,q)}$ (Exercise 9, c)). Prove that the cross product $A[\mathscr{E},L]$ of L and the extension $\mathscr{E}$ associated with $c_{\theta}$ is isomorphic to the quotient of the K-algebra $L[X]_{\sigma}$ (VIII, p. 11) by the (two-sided) ideal generated by the central element $X^n-\theta$. The class of $A_{\theta}$ in Br(L$/K$) corresponds to the class of $\theta$ (mod $N_{L/K}(L^*)$) by the isomorphism defined above.