Proceedings of the Japan Academy, Series A, Mathematical Sciences

Explicit lifts of quintic Jacobi sums and period polynomials for $\mathbf {F}_{ {q}}$

Akinari Hoshi

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Abstract

In this paper, we construct explicit lifts of quintic Jacobi sums for finite fields via integer solutions of Dickson's system. Namely we give a procedure to compute quintic Jacobi sums for extended field $\mathbf{F}_{p^{s+t}}$ by using quintic Jacobi sums for $\mathbf{F}_{p^s}$ and for $\mathbf{F}_{p^t}$. We also have the multiplication formula from $\mathbf{F}_{p^s}$ to $\mathbf{F}_{p^{ns}}$ as a special case. By the quintuplication formula, we obtain the explicit factorization of the quintic period polynomials for finite fields.

Article information

Source
Proc. Japan Acad. Ser. A Math. Sci., Volume 82, Number 7 (2006), 87-92.

Dates
First available in Project Euclid: 10 October 2006

Permanent link to this document
https://projecteuclid.org/euclid.pja/1160485503

Digital Object Identifier
doi:10.3792/pjaa.82.87

Mathematical Reviews number (MathSciNet)
MR2265605

Zentralblatt MATH identifier
1116.11093

Subjects
Primary: 11E25: Sums of squares and representations by other particular quadratic forms 11L05: Gauss and Kloosterman sums; generalizations 11T22: Cyclotomy 11T24: Other character sums and Gauss sums

Keywords
Jacobi sums Gaussian periods Dickson's system Gauss sums

Citation

Hoshi, Akinari. Explicit lifts of quintic Jacobi sums and period polynomials for $\mathbf {F}_{ {q}}$. Proc. Japan Acad. Ser. A Math. Sci. 82 (2006), no. 7, 87--92. doi:10.3792/pjaa.82.87. https://projecteuclid.org/euclid.pja/1160485503


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