×

zbMATH — the first resource for mathematics

Congruences for Brewer sums. (English) Zbl 1107.11032
The Brewer sums considered here are \[ \Lambda_n(a)=\sum_{x=0}^{p-1} L(D_n(x,a),p), \] where \(p\) is a prime, \(L\) is the Legendre symbol and \(D_n(x,a)\) is the \(n\)th order Dickson polynomial of the first kind. \(\Lambda_n(a)\) is easily seen to be 0 when \((n, p^2-1)=1\) or \(p\equiv 3\pmod{4}\). The author considers the case where \(p\equiv 1\pmod{4}\), \(n\) is an odd prime dividing \(p^2-1\) and determines which \(\Lambda_n(a)\) are 0. The proof uses explicit factorizations of the Dickson polynomials over finite fields.

MSC:
11L10 Jacobsthal and Brewer sums; other complete character sums
11T06 Polynomials over finite fields
12Y05 Computational aspects of field theory and polynomials (MSC2010)
PDF BibTeX Cite
Full Text: DOI
References:
[1] Berndt, B.C.; Evans, R.J.; Williams, K.S., Gauss and Jacobi sums, (1998), Wiley New York · Zbl 0906.11001
[2] Bhargava, M.; Zieve, M.E., Factoring dickson polynomials over finite fields, Finite fields appl., 5, 103-111, (1999) · Zbl 0929.11060
[3] Brewer, B.W., On certain character sums, Trans. amer. math. soc., 99, 241-245, (1961) · Zbl 0103.03205
[4] Brewer, B.W., On primes of the form \(u^2 + 5 v^2\), Proc. amer. math. soc., 17, 502-509, (1966) · Zbl 0147.29801
[5] Chou, W.-S., The factorization of dickson polynomials over finite fields, Finite fields appl., 3, 84-96, (1997) · Zbl 0910.11052
[6] Leprévost, F.; Morain, F., Revêtements de courbes elliptiques à multiplication complexe par des courbes hyperelliptiques et sommes de caractères, J. number theory, 64, 165-182, (1997) · Zbl 0874.11044
[7] R. Lidl, G.L. Mullen, G. Turnwald, Dickson Polynomials, Pitman Monograph, vol. 65, Longman Sci. Tech., Harlow, UK, 1993.
[8] Lidl, R.; Niederreiter, H., Finite fields, (1997), Cambridge University Press Cambridge
[9] Williams, K.S., Note on Dickson’s permutation polynomials, Duke math. J., 38, 659-665, (1971) · Zbl 0235.12011
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.