# Primitive character, coductor and Jacobi sum

Dirichlet characters

Recall that is a character $\chi$ of modulo $q$ is said to be induced by a character $\chi'$ of modulo $d$ if $\chi(n)=\chi'(n)$ for every $n\in\mathbf{Z}$ with $\gcd(n,q)=1$, here $d$ is a divisor of $q$.

### Jacobi sum

Jacobi sum is a type of character sum formed with Dirichlet characters. The Jacobi sms for Dirichlet characters $\chi,\chi'$ modulo a prime number $p$, defined by

$\displaystyle J(\chi,\chi')=\sum_{n\in \mathbf{Z}/p\mathbf{Z}} \chi(n)\chi(1-n),$

Jacobi sums are the analogues for finite fields of the beta function.