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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Some arithmetic properties of the minimal polynomials of Gauss sums


Author: Da Qing Wan
Journal: Proc. Amer. Math. Soc. 100 (1987), 225-228
MSC: Primary 11L05
MathSciNet review: 884455
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Abstract: For the minimal polynomial $ f(x) = {x^k} + {b_1}{x^{k - 1}} + \cdots + {b_k}$ of $ \sum\nolimits_{n = 0}^{p - 1} {\exp (2\pi i{n^k}/p)} $ over $ Q$, where $ p$ is a $ \operatorname{prime} \equiv 1(\bmod k)$, we evaluate $ {b_1},{b_2}$ and prove $ \left. p \right\vert{b_i}(i = 1, \ldots ,k)$ but $ {p^2}\nmid {b_j}(j = 2,k)$. Also, we raise the interesting conjecture that $ {p^2}\nmid {b_j}$ for $ k \geq j \geq 2$.


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DOI: https://doi.org/10.1090/S0002-9939-1987-0884455-8
Article copyright: © Copyright 1987 American Mathematical Society