On the coefficients of the minimal polynomials of Gaussian periods
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- by S. Gupta and D. Zagier PDF
- Math. Comp. 60 (1993), 385-398 Request permission
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.References
- S. Gurak, Minimal polynomials for Gauss circulants and cyclotomic units, Pacific J. Math. 102 (1982), no. 2, 347–353. MR 686555
- S. Gurak, Minimal polynomials for circular numbers, Pacific J. Math. 112 (1984), no. 2, 313–331. MR 743988
- D. H. Lehmer and Emma Lehmer, Cyclotomy with short periods, Math. Comp. 41 (1983), no. 164, 743–758. MR 717718, DOI 10.1090/S0025-5718-1983-0717718-1
Additional Information
- © Copyright 1993 American Mathematical Society
- Journal: Math. Comp. 60 (1993), 385-398
- MSC: Primary 11L05; Secondary 11T22, 11Y40
- DOI: https://doi.org/10.1090/S0025-5718-1993-1155574-7
- MathSciNet review: 1155574