Some Results on Gauss Sums

is again a Dirichlet character modulo n. In fact, the set of Dirichlet characters modulo n is again a multiplicative group, called the dual group of (Z/nZ)× and denoted ̂ (Z/nZ)×. The identity element of the dual group, mapping every element of (Z/nZ)× to 1, is the trivial character modulo n, denoted 1n or just 1 when n is clear, Since (Z/nZ)× is a finite group the values taken by any Dirichlet character are complex roots of unity, specifically φ(n)th roots of unity. One consequence of this is that there exist only finitely many Dirichlet characters modulo any given positive integer n. Another consequences is that the inverse of a Dirichlet character is its complex conjugate, defined by the rule