Multiplicative character of a finite field ($\chi :{\mathbb{F}}_q^{\times }\to {\mathbb{C}}^{\times }$)

A multiplicative character is a group homomorphism from a finite field's multiplicative group of a finite field to the complex unit circle. A generator of the cyclic multiplicative group of a finite field produces a character of every order dividing $q-1$. Extend every such character to zero by value zero, including the trivial multiplicative character, when using cyclotomic indicator formulas.

When $\chi ,\eta ,\chi \eta$ are nontrivial, changing variables in a product of Gauss sums of finite-field characters gives $G(\chi )G(\eta )=J(\chi ,\eta )G(\chi \eta )$, hence $|J|=\sqrt{q}$. If at least one character is nontrivial and their product is trivial, or the other character is trivial, the absolute value is one. Thus all non-main terms in a two-character cyclotomic intersection count are $O(\sqrt{q})$.

Here $\psi$ is a nontrivial additive character of the finite field. For nontrivial $\chi$, substituting $x=ty$ in the squared absolute value givesThe inner sum is $q-1$ at $t=1$ and $-1$ otherwise; multiplicative character orthogonality completes the identity.