Dirichlet character

A Dirichlet characterDlmfMathworldPlanetmath modulo m is a group homomorphismMathworldPlanetmath from (m)* to *. Dirichlet characters are usually denoted by the Greek letter χ. The functionMathworldPlanetmath

γ(n)={χ(nmodm),if gcd(n,m)=1,0,if gcd(n,m)>1.

is also referred to as a Dirichlet character. The Dirichlet characters modulo m form a group if one defines χχ to be the function which takes a(m)* to χ(a)χ(a). It turns out that this resulting group is isomorphic to (m)*. The trivial character is given by χ(a)=1 for all a(m)*, and it acts as the identity elementMathworldPlanetmath for the group. A characterPlanetmathPlanetmath χ modulo m is said to be induced by a character χ modulo m if mm and χ(n)=χ(nmodm). A character which is not induced by any other character is called primitive. If χ is non-primitive, the gcd of all such m is called the conductor of χ.


  • Legendre symbolDlmfMathworldPlanetmath (np) is a Dirichlet character modulo p for any odd prime p. More generally, Jacobi symbolDlmfMathworldPlanetmath (nm) is a Dirichlet character modulo m.

  • The character modulo 4 given by χ(1)=1 and χ(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