Relationship of Gaussian Periods and Gauss Sums
The Gaussian periods are related to the Gauss sums for which the character χ is trivial on H. Such χ take the same value at all elements a in a fixed coset of H in G. For example, the quadratic character mod p described above takes the value 1 at each quadratic residue, and takes the value -1 at each quadratic non-residue. The Gauss sum can thus be written as a linear combination of Gaussian periods (with coefficients χ(a)); the converse is also true, as a consequence of the orthogonality relations for the group (Z/nZ)×. In other words, the Gaussian periods and Gauss sums are each other's Fourier transforms. The Gaussian periods generally lie in smaller fields, since for example when n is a prime p, the values χ(a) are (p − 1)-th roots of unity. On the other hand, Gauss sums have nicer algebraic properties.
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