On power integral bases of unramified cyclic extensions of prime degree (Q1840609)

Let \(p\) be a prime, \(K\) a number field containing a primitive \(p\)-th root of unity \(\zeta_p\), and \(L/K\) a cyclic extension of degree \(p\) with Galois group \(G\). Then the author shows that the following two conditions are equivalent: (1) \(L/K\) is unramified, and \(O_L = O_K[\alpha]\) for some \(\alpha \in O_L\) with \(\alpha^\sigma - \zeta_p\alpha \in O_K\) for some \(\sigma \in G.\) (2) There is a primary unit \(\varepsilon \in E_K\) such that \(L = K(\varepsilon^{1/p})\). As the author remarks, this result was independently obtained (using different methods) by F.~Kawamoto and N.~Suwa.