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Mathematics of Computation

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Jacobi sums and new families of irreducible polynomials of Gaussian periods

Author: F. Thaine
Journal: Math. Comp. 70 (2001), 1617-1640
MSC (2000): Primary 11R18, 11R21, 11T22
Published electronically: May 11, 2001
MathSciNet review: 1836923
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Let $m> 2$, $\zeta _m $ an $m$-th primitive root of 1, $q\equiv 1$ mod $2m$ a prime number, $s=s_{q}$ a primitive root modulo $q$ and $f=f_{q}=(q-1)/m$. We study the Jacobi sums $J_{a,b}=-\sum _{k=2}^{q-1}\zeta _m ^{\, a\, \text{ind}_{s}(k)+b\, \text{ind}_{s}(1-k)}$, $0\leq a, b\leq m-1$, where $\text{ind}_{s}(k)$ is the least nonnegative integer such that $s^{\, \text{ind}_{s}(k)}\equiv k$ mod $q$. We exhibit a set of properties that characterize these sums, some congruences they satisfy, and a MAPLE program to calculate them. Then we use those results to show how one can construct families $P_{q}(x)$, $q\in \mathcal{P}$, of irreducible polynomials of Gaussian periods, $\eta _{i}=\sum _{j=0}^{f-1}\zeta _q^{s^{i+mj}}$, of degree $m$, where $\mathcal{P}$ is a suitable set of primes $\equiv 1$ mod $2m$. We exhibit examples of such families for several small values of $m$, and give a MAPLE program to construct more of them.

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Additional Information

F. Thaine
Affiliation: Department of Mathematics and Statistics - CICMA, Concordia University, 1455, de Maisonneuve Blvd. W., Montreal, Quebec, H3G 1M8, Canada

Received by editor(s): September 15, 1998
Received by editor(s) in revised form: January 19, 2000
Published electronically: May 11, 2001
Additional Notes: This work was supported in part by grants from NSERC and FCAR
Article copyright: © Copyright 2001 American Mathematical Society