Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
5772826 | Journal of Pure and Applied Algebra | 2017 | 12 Pages |
Abstract
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).
Related Topics
Physical Sciences and Engineering
Mathematics
Algebra and Number Theory
Authors
Tae Maeda, Anri Nakaya, Kaori Ota,