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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.

##### MSC:
 11L10 Jacobsthal and Brewer sums; other complete character sums 11T06 Polynomials over finite fields 12Y05 Computational aspects of field theory and polynomials (MSC2010)
##### Keywords:
Brewer sums; Dickson polynomials; finite fields
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##### References:
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