In mathematics, a multiplicative character (or linear character, or simply character) on a group ''G'' is a group homomorphism from ''G'' to the multiplicative group of a field , usually the field of

complex numbers In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the for ...

. If ''G'' is any group, then the set Ch(''G'') of these morphisms forms an

abelian group In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is com ...

under pointwise multiplication. This group is referred to as the character group of ''G''. Sometimes only ''unitary'' characters are considered (characters whose

image An image is a visual representation of something. It can be two-dimensional, three-dimensional, or somehow otherwise feed into the visual system to convey information. An image can be an artifact, such as a photograph or other two-dimensio ...

is in the

unit circle In mathematics, a unit circle is a circle of unit radius—that is, a radius of 1. Frequently, especially in trigonometry, the unit circle is the circle of radius 1 centered at the origin (0, 0) in the Cartesian coordinate system in the Eucli ...

); other such homomorphisms are then called ''quasi-characters''. Dirichlet characters can be seen as a special case of this definition. Multiplicative characters are

linearly independent In the theory of vector spaces, a set of vectors is said to be if there is a nontrivial linear combination of the vectors that equals the zero vector. If no such linear combination exists, then the vectors are said to be . These concepts ...

, i.e. if

\chi_1, \chi_2, \ldots, \chi_n

are different characters on a group ''G'' then from

a_1\chi_1 + a_2\chi_2 + \cdots + a_n\chi_n = 0

it follows that

a_1 = a_2 = \cdots = a_n = 0.

Examples

*Consider the (''ax'' + ''b'')-group ::

G := \left\.

: Functions ''f''_''u'' : ''G'' → C such that

f_u \left(\begin
a & b \\
0 & 1  \end\right)=a^u,

where ''u'' ranges over complex numbers C are multiplicative characters. * Consider the multiplicative group of positive

real numbers In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...

(R⁺,·). Then functions ''f''_''u'' : (R⁺,·) → C such that ''f''_''u''(''a'') = ''a''^''u'', where ''a'' is an element of (R⁺, ·) and ''u'' ranges over complex numbers C, are multiplicative characters.

References

* {{citation, title=Galois Theory, series=Notre Dame Mathematical Lectures, number 2, authorlink=Emil Artin, first=Emil, last= Artin, year=1966, publisher = Arthur Norton Milgram (Reprinted Dover Publications, 1997), isbn=978-0-486-62342-9 Lectures Delivered at the University of Notre Dame Group theory