Abstract
We obtain a reduction formula for an imprimitive Gaussian sum with a numerical character in an algebraic number field, i.e. a formula that expresses that sum as a product of several elementary factors times a primitive, proper, normed Gaussian sum (formulae (16) and (19)). We also introduce Gaussian sums with Hecke characters and derive a similar reduction formula for them. The derivation is based on an inversion formula for a multivariable theta function associated with the number field, twisted with the numerical character.
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