Indices of Kummer extensions of prime degree

Let p be a prime and let ζp be a primitive p-th root of unity. For a finite extension k of Q containing ζp, we consider a Kummer extension L/k of degree p. In this paper, we show that if k=Q(ζp) and the class number of k is one, the index of L/k is one. We also show that if L/k is tamely ramified wi...

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Bibliographic Details
Published in:Journal of pure and applied algebra 2017-11, Vol.221 (11), p.2815-2826
Main Authors: Maeda, Tae, Nakaya, Anri, Ota, Kaori
Format: Article
Language:English
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Summary:Let p be a prime and let ζp be a primitive p-th root of unity. For a finite extension k of Q containing ζp, we consider a Kummer extension L/k of degree p. In this paper, we show that if k=Q(ζp) and the class number of k is one, the index of L/k is one. We also show that if L/k is tamely ramified with a normal integral basis, the index is at most a power of p. In the last section, we show that there exist infinitely many cubic Kummer extensions of Q(ζ3) for both wildly and tamely ramified cases, whose integer rings do not have a power integral basis over that of Q(ζ3).
ISSN:0022-4049
1873-1376
DOI:10.1016/j.jpaa.2017.02.001