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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.

11L10 Jacobsthal and Brewer sums; other complete character sums
11T06 Polynomials over finite fields
12Y05 Computational aspects of field theory and polynomials (MSC2010)
Full Text: DOI
[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
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