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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

Generalized cyclotomic periods


Author: Ronald J. Evans
Journal: Proc. Amer. Math. Soc. 81 (1981), 207-212
MSC: Primary 10G05
MathSciNet review: 593458
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Abstract: Let $ n$ and $ q$ be relatively prime integers with $ n > 1$, and set $ N$ equal to twice the product of the distinct prime factors of $ n$. Let $ t(n)$ denote the order of $ q$ $ (\bmod n)$. Write $ \eta = \sum _{\upsilon = 0}^{t(n) - 1}{a_\upsilon }\zeta _n^{{q^\upsilon }}$ where $ {\zeta _n} = \exp (2\pi i/n)$. If $ {a_\upsilon } = 1$ for all $ \upsilon $, then $ \eta $ is Kummer's cyclotomic period, and if $ {a_\upsilon } = \exp (2\pi i\upsilon /t(n))$ for each $ \upsilon $, then $ \eta $ is a type of Lagrange resolvent. For certain classes of $ {a_\upsilon } \in {\mathbf{Q}}(\zeta _n^N)$, necessary and sufficient conditions for the vanishing of $ \eta $ are given, and the degree of $ \eta $ over $ {\mathbf{Q}}$ is determined.


References [Enhancements On Off] (What's this?)

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1981-0593458-7
PII: S 0002-9939(1981)0593458-7
Keywords: Cyclotomic periods
Article copyright: © Copyright 1981 American Mathematical Society