Editing Character (mathematics) (section)

== Multiplicative character ==
{{main|multiplicative character}}

A '''multiplicative character''' (or '''linear character''', or simply '''character''') on a group ''G'' is a [[group homomorphism]] from ''G'' to the [[unit group|multiplicative group]] of a field {{Harv|Artin|1966}}, usually the field of [[complex number]]s.  If ''G'' is any group, then the set Ch(''G'') of these morphisms forms an [[abelian group]] under pointwise multiplication.

This group is referred to as the [[character group]] of ''G''. Sometimes only ''unitary'' characters are considered (thus the image is in the [[unit circle]]); other such homomorphisms are then called ''quasi-characters''. [[Dirichlet character]]s can be seen as a special case of this definition.

Multiplicative characters are [[linear independence|linearly independent]], i.e. if <math>\chi_1,\chi_2, \ldots , \chi_n </math> are different characters on a group ''G'' then from <math>a_1\chi_1+a_2\chi_2 + \dots + a_n \chi_n = 0 </math> it follows that <math>a_1=a_2=\cdots=a_n=0 </math>.