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Generalized Kummer theory and its applications. (English) Zbl 1188.11054

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)], 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 \(3\).

MSC:

11R20 Other abelian and metabelian extensions
12E10 Special polynomials in general fields
11R16 Cubic and quartic extensions
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References:

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