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Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)



On the coefficients of the minimal polynomials of Gaussian periods

Authors: S. Gupta and D. Zagier
Journal: Math. Comp. 60 (1993), 385-398
MSC: Primary 11L05; Secondary 11T22, 11Y40
MathSciNet review: 1155574
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Abstract: Let l be a prime number and m a divisor of $ l - 1$. Then the Gauss period $ \omega = \zeta + {\zeta ^\lambda } + {\zeta ^{{\lambda ^2}}} + \cdots + {\zeta ^{{\lambda ^{m - 1}}}}$ where $ \zeta = {e^{2\pi i/l}}$ and $ \lambda $ is a primitive mth root of unity modulo l, generates a subfield K of $ \mathbb{Q}(\zeta )$ of degree $ (l - 1)/m$. In this paper we study the reciprocal minimal polynomial $ {F_{l,m}}(X) = {N_{K/\mathbb{Q}}}(1 - \omega X)$ of $ \omega $ over $ \mathbb{Q}$. It will be shown that for fixed m and every N we have $ {F_{l,m}}(X) \equiv {({B_m}{(X)^l}/(1 = mX))^{1/m}}\;\pmod {{X^N}}$ for all but finitely many "exceptional primes" l (depending on m and N), where $ {B_m}(X) \in \mathbb{Z}[[X]]$ is a power series depending only on m . A method of computation of this set of exceptional primes is presented. The generalization of the results to the case of composite l is also discussed.

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Article copyright: © Copyright 1993 American Mathematical Society