Gauss sums over finite fields and roots of unity

Oliver, Robert

doi:10.1090/S0002-9939-2010-10662-7

Gauss sums over finite fields and roots of unity
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by Robert J. Lemke Oliver

Proc. Amer. Math. Soc. 139 (2011), 1273-1276

DOI: https://doi.org/10.1090/S0002-9939-2010-10662-7

Published electronically: September 30, 2010

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Abstract:

Let $\chi$ be a non-trivial character of $\mathbb {F}_{q}^\times$, and let $g(\chi )$ be its associated Gauss sum. It is well known that $g(\chi )=\varepsilon (\chi )\sqrt {q}$, where $|\varepsilon (\chi )|=1$. Using the $p$-adic gamma function, we give a new proof of a result of Evans which gives necessary and sufficient conditions for $\varepsilon (\chi )$ to be a root of unity.

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