Explicit formulas for Hecke Gauss sums in quadratic number fields

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We derive an explicit formula for Hecke Gauss sums of quadratic number fields. As an immediate consequence we obtain a quadratic reciprocity law in quadratic number fields which generalizes the classical one given by Hecke. The proofs use, apart from the well-known formulas for ordinary Gauss sums, only elementary algebraic manipulations.

Communicated by U. Kühn.
During the preparation of this article the first author was supported by TÜBİTAK, the Scientific and Technological Research Council of Turkey, and the University of Siegen.