Generalized Kummer theory and its applications (Q1017372)

Let \(n\) be an odd natural number greater than \(2\) and \(\zeta\) a primitive \(n\)th root of unity. Let \(k=\mathbb{Q}(\zeta + \zeta^{-1})\). In [Proc. Am. Math. Soc. 130, No. 8, 2215--2218 (2002; Zbl 0990.12005)], \textit{Y. Rikuna} defined a polynomial \(P\) which is generic over the field \(k\) for the cyclic group of order \(n\), and obtain a generalized Kummer theory. In the paper under review, the author recalls results, coming from the use of \(P\), concerning ramification and Artin symbols from his papers [Manuscr. Math. 114, No. 3, 265--279 (2004; Zbl 1093.11068) and Tokyo J. Math. 30, No. 1, 169--178 (2007; Zbl 1188.11053)], and applies them when the degree of \(P\) is \(3\).