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NUMBER-THEORETIC CHARACTERS If ''G'' is Group , a character is a Group Homomorphism into the Multiplicative Group of a field (as defined in Emil Artin's book on Galois Theory), usually the field of Complex Numbers . If ''A'' is an Abelian Group , then the set Ch(''A'') of these morphisms forms a group under the operation :χaχb=χab. This group is referred to as the Character Group . Sometimes only ''unitary'' characters are considered (so that the image is in the Unit Circle ); other such homomorphisms are then called ''quasi-characters''. Dirichlet Character s can be seen a special case of this definition. REPRESENTATION CHARACTERS If ''f'' is a finite-dimensional Representation of a Group ''G'', then the character of the representation is the function from ''G'' to the complex numbers given by the Trace of ''f''. In general, the trace is neither a group homomorphism, nor does the set of traces form a group. The study of representations by means of their characters is called Character Theory . ALGEBRAIC CHARACTERS
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