In
number theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic functions. Number theorists study prime numbers as well as the properties of mathematical objects constructed from integers (for example ...
, a genus character of a
quadratic number field
In algebraic number theory, a quadratic field is an algebraic number field of degree two over \mathbf, the rational numbers.
Every such quadratic field is some \mathbf(\sqrt) where d is a (uniquely defined) square-free integer different from 0 an ...
''K'' is a character of the
genus group of ''K''. In other words, it is a real character of the
narrow class group In algebraic number theory, the narrow class group of a number field ''K'' is a refinement of the class group of ''K'' that takes into account some information about embeddings of ''K'' into the field of real numbers.
Formal definition
Suppose ...
of ''K''. Reinterpreting this using the
Artin map The Artin reciprocity law, which was established by Emil Artin in a series of papers (1924; 1927; 1930), is a general theorem in number theory that forms a central part of global class field theory. The term "reciprocity law" refers to a long line ...
, the collection of genus characters can also be thought of as the unramified real characters of the
absolute Galois group
In mathematics, the absolute Galois group ''GK'' of a field ''K'' is the Galois group of ''K''sep over ''K'', where ''K''sep is a separable closure of ''K''. Alternatively it is the group of all automorphisms of the algebraic closure of ''K'' ...
of ''K'' (i.e. the characters that factor through the Galois group of the
genus field of ''K'').
References
*Chapter II of
*
*Section 12.5 of
*Section 2.3 of
Algebraic number theory
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