Abstract
From a Davenport-Hasse identity of Gauss sums an identity of a hyper-Kloosterman sum has been deduced. Using this identity the theory of Kloosterman sheaves and equidistribution of hyper-Kloosterman sums can be applied to an exponential sum over a cyclic algebraic number field of prime degree. This identity might also be applied to base change problems in representation theory via a possible relative trace formula over the cyclic number field.
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