# Genus character

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{{Short description|Concept in number theory}}
In [number theory](/source/number_theory), a '''genus character''' of a [quadratic number field](/source/quadratic_number_field) ''K'' is a character of the [genus group](/source/genus_group) of ''K''. In other words, it is a real character of the [narrow class group](/source/narrow_class_group) of ''K''. Reinterpreting this using the [Artin map](/source/Artin_map), the collection of genus characters can also be thought of as the unramified real characters of the [absolute Galois group](/source/absolute_Galois_group) of ''K'' (i.e. the characters that factor through the Galois group of the [genus field](/source/genus_field) of ''K'').

==References==

*Chapter II of {{Citation
| last=Siegel
| first=Carl
| title=Advanced Analytic Number Theory
}}
*{{Citation | last1=Bertolini | first1=Massimo | last2=Darmon | first2=Henri | title=  The rationality of Stark-Heegner points over genus fields of real quadratic fields | year=2009 | doi=10.4007/annals.2009.170.343  |mr=2521118 | journal=[Annals of Mathematics](/source/Annals_of_Mathematics) | issn=0003-486X | volume=170 | pages=343–369| doi-access=free }}
*Section 12.5 of {{Citation
| last=Iwaniec
| first=Henryk
| title=Topics in classical automorphic forms
}}
*Section 2.3 of {{Citation
| last=Lemmermeyer
| first=Franz
| title=Reciprocity laws: From Euler to Eisenstein
}}

Category:Algebraic number theory

{{numtheory-stub}}

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