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associated to a ray class group by class field theory, and its Galois group is isomorphic to the corresponding ray class group. The proof of existence of a ray class field of a given ray class group is long and indirect and there is in general no known easy way to construct it (though explicit
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The term "ray class group" is a translation of the German term "Strahlklassengruppe". Here "Strahl" is German for ray, and often means the positive real line, which appears in the positivity conditions defining ray class groups.
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The ray class groups defined using ideles are naturally isomorphic to those defined using ideals. They are sometimes easier to handle theoretically because they are all quotients of a single group, and thus easier to compare.
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Weber introduced ray class groups in 1897. Takagi proved the existence of the corresponding ray class fields in about 1920. Chevalley reformulated the definition of ray class groups in terms of ideles in 1933.
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59:, p.6) uses "Strahl" to mean a certain group of ideals defined using positivity conditions, and uses "Strahlklasse" to mean a coset of this group.
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is the ray class field corresponding to the unit ideal and the set of all real places, so it is the smallest narrow ray class field.
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is the ray class field corresponding to the unit ideal and the empty set of real places, so it is the smallest ray class field. The
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There are two slightly different notions of what a ray class field is, as authors differ in how the infinite primes are treated.
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51:. Every finite abelian extension of a number field is contained in one of its ray class fields.
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constructions are known in some special cases such as imaginary quadratic fields).
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213:. Some authors use the term "ray class group" to mean "narrow ray class group".
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of real places as the quotient of the idele class group by image of the group
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The ray class field of a ray class group is the (unique) abelian extension
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is restricted to be totally positive, the group is called the
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Some authors use a more general definition, where the group
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is a subset of the real places, then the ray class group of
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568:{\displaystyle \mathbb {Q} (\cos({\frac {2\pi }{m}}))}
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is allowed to be all nonzero real numbers for certain
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Jahresbericht der
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Chevalley redefined the ray class group of an ideal
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489:, then the ray class group of (
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446:{\displaystyle \prod U_{p}\,}
268:{\displaystyle \prod U_{p}\,}
229:Ray class fields using ideles
189: ≡ 1 mod
148:{\displaystyle I^{m}/P^{m}\,}
75:Ray class fields using ideals
516:. The ray class field for (
453:in the idele class group of
220:is the abelian extension of
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618:Algebraische Zahlentheorie
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237:and a set
551:π
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430:∏
369:dividing
357:to 1 mod
355:congruent
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665:Category
616:(1999).
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167:co-prime
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197:. When
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66:History
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