Dirichlet character

A Dirichlet character modulo $m$ is a group homomorphism from ${\left(\frac{\mathbb{Z}}{m\mathbb{Z}}\right)}^{*}$ to ${ℂ}^{*}$. Dirichlet characters are usually denoted by the Greek letter $\chi$. The function

is also referred to as a Dirichlet character. The Dirichlet characters modulo $m$ form a group if one defines $\chi {\chi }^{\prime }$ to be the function which takes $a\in {\left(\frac{\mathbb{Z}}{m\mathbb{Z}}\right)}^{*}$ to $\chi (a){\chi }^{\prime }(a)$. It turns out that this resulting group is isomorphic to ${\left(\frac{\mathbb{Z}}{m\mathbb{Z}}\right)}^{*}$. The trivial character is given by $\chi (a)=1$ for all $a\in {\left(\frac{\mathbb{Z}}{m\mathbb{Z}}\right)}^{*}$, and it acts as the identity element for the group. A character $\chi$ modulo $m$ is said to be induced by a character ${\chi }^{\prime }$ modulo $m^{\prime }$ if $m^{\prime }\mid m$ and $\chi (n)={\chi }^{\prime }(nmod{m}^{\prime })$. A character which is not induced by any other character is called primitive. If $\chi$ is non-primitive, the $\gcd$ of all such $m^{\prime }$ is called the conductor of $\chi$.

Examples:

• Legendre symbol $\left(\frac{n}{p}\right)$ is a Dirichlet character modulo $p$ for any odd prime $p$. More generally, Jacobi symbol $\left(\frac{n}{m}\right)$ is a Dirichlet character modulo $m$.

• The character modulo $4$ given by $\chi (1)=1$ and $\chi (3)=-1$ is a primitive character modulo $4$. The only other character modulo $4$ is the trivial character.

Title	Dirichlet character
Canonical name	DirichletCharacter
Date of creation	2013-03-22 13:22:31
Last modified on	2013-03-22 13:22:31
Owner	bbukh (348)
Last modified by	bbukh (348)
Numerical id	10
Author	bbukh (348)
Entry type	Definition
Classification	msc 11A25
Related topic	CharacterOfAFiniteGroup
Defines	trivial character
Defines	primitive character
Defines	conductor
Defines	induced character