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On character sums in finite fields. (English) Zbl 0021.20202

##### Keywords:
Number fields, function fields
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 [1] A normalised polynomial is one in which the coefficient of the highest power ofx is I. [2] Journal für Math. (Crelle), 172 (1934), 37–54. [3] (f,g), where $$\Phi$$ runs through the roots ofg(x)=0. [4] Forv=0, there is only one polynomialg, namely 1. Also (f i , 1)=1. Hence $$\sigma$$o=I. [5] This was conjectured by Hasse (loc. cit., 52). A proof different from that in the present paper has been given (in an unpublished MS) by Witt. [6] Hamburg Abh. 10 (1934), 325–348. [7] ForK>3, all previously known inequalities forS(f, $$\chi$$) dealt only with the case in which all the characters are quadratic. For an account of them, see Davenport, Journal London Math. Soc. 8 (1933), 46–52. They are all weaker than (13) above. [8] Math. Annalen 114 (1937), 476–492. [9] loc. cit. Math. Annalen 114 (1937), 476–492 (20). [10] Journal für Math. (Crelle), 172 (1934), 151–182. The Gaussian sums are defined there with a negative sign prefixed. [11] Journal für Math. (Crelle), 176 (1937), 189–191. [12] Ifv=k, the first line of (23) is empty. [13] Ifv=K, the first line of (27) is empty. [14] This proof is essentially the same as one given in a previous paper (Journal London Math. Soc. 7 (1932), 117–121). [15] The proof is a refinement and extension of a method previously used in connection with a special case of the problem (see Quarterly Journal of Math. 8 (1937), 308–312). [16] See Davenport, Journal London Math. Soc. 8 (1933), 46–52. · Zbl 0006.24901 · doi:10.1112/jlms/s1-8.1.46
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