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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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

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

First available in Project Euclid: 10 October 2006

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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

Jacobi sums Gaussian periods Dickson's system Gauss sums


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.

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  • V. V. Acharya and S. A. Katre, Cyclotomic numbers of order $2l$, $l$ an odd prime, Acta Arith. 69 (1995), 51–74.
  • N. Anuradha and S. A. Katre, Number of points on the projective curves $aY^l=bX^l+cZ^l$ and $aY^{2l}=bX^{2l}+cZ^{2l}$ defined over finite fields, $l$ an odd prime, J. Number Theory 77 (1999), 288–313.
  • L. D. Baumert, W. H. Mills and R. L. Ward, Uniform cyclotomy, J. Number Theory 14 (1982), 67–82.
  • B. C. Berndt, R. J. Evans and K. S. Williams, Gauss and Jacobi sums, Canadian Mathematical Society Series of Monographs and Advanced Texts, Wiley, New York, 1998.
  • L. E. Dickson, Cyclotomy, higher congruences and Waring's problem, Amer. J. Math. 57 (1935), 391–424.
  • R. J. Evans, Pure Gauss sums over finite fields, Mathematika. 28 (1981), 239–248.
  • C. F. Gauss, Disquisitiones Arithmeticae, Section 358.
  • S. Gurak, Period polynomials for $F\sb {p\sp 2}$ of fixed small degree, in Finite fields and applications (Augsburg, 1999) 196–207, Springer, Berlin.
  • S. Gurak, Period polynomials for $ \mathbf{F}\sb q$ of fixed small degree, in Number theory, 127–145, Amer. Math. Soc., Providence, 2004.
  • K. Hashimoto and A. Hoshi, Families of cyclic polynomials obtained from geometric generalization of Gaussian period relations, Math. Comp. 74 (2005), 1519–1530.
  • K. Hashimoto and A. Hoshi, Geometric generalization of Gaussian period relations with application to Noether's problem for meta-cyclic groups, Tokyo J. Math. 28 (2005), 13–32.
  • A. Hoshi, Multiplicative quadratic forms on algebraic varieties, Proc. Japan Acad. Ser. A Math. Sci. 79 (2003), no.4, 71–75.
  • S. A. Katre and A. R. Rajwade, Unique determination of cyclotomic numbers of order five, Manuscripta Math. 53 (1985), 65–75.
  • S. A. Katre and A. R. Rajwade, Complete solution of the cyclotomic problem in $ \mathbf{F}_q^{*}$ for any prime modulus $l$, $q=p^\alpha$, $p\equiv 1$ (mod $l$), Acta Arith. 45 (1985), 183–199.
  • G. Myerson, Period polynomials and Gauss sums for finite fields, Acta Arith. 39 (1981), 251–264.
  • F. Thaine, Properties that characterize Gaussian periods and cyclotomic numbers, Proc. Amer. Math. Soc. 124 (1996), 35–45.
  • F. Thaine, On the coefficients of Jacobi sums in prime cyclotomic fields, Trans. Amer. Math. Soc. 351 (1999), 4769–4790.
  • F. Thaine, Families of irreducible polynomials of Gaussian periods and matrices of cyclotomic numbers, Math. Comp. 69 (2000), 1653–1666.
  • F. Thaine, Jacobi sums and new families of irreducible polynomials of Gaussian periods, Math. Comp. 70 (2001), 1617–1640.
  • F. Thaine, On Gaussian periods that are rational integers, Michigan Math. J. 50 (2002), 313–337.
  • F. Thaine, Cyclic polynomials and the multiplication matrices of their roots, J. Pure Appl. Algebra 188 (2004), 247–286.
  • P. V. Wamelen, Jacobi sums over finite fields, Acta Arith. 102 (2002), 1–20.
  • S. Wolfram, The Mathematica book, Fourth ed., Wolfram Media, Inc., Cambridge Univ. Press, Cambridge-New York, 1999.