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

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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 A proof different from that in the present paper has been given (in an unpublished MS) by Witt.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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