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 A proof different from that in the present paper has been given (in an unpublished MS) by Witt.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