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| In [[algebraic number theory]], the '''Hilbert class field''' ''E'' of a [[number field]] ''K'' is the [[Maximal abelian extension|maximal abelian]] [[unramified]] extension of ''K''. Its degree over ''K'' equals the class number of ''K'' and the [[Galois group]] of ''E'' over ''K'' is canonically isomorphic to the [[ideal class group]] of ''K'' using [[Frobenius element]]s for [[prime ideal]]s in ''K''.
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| In this context, the Hilbert class field of ''K'' is not just unramified at the [[Prime_number#Primes_in_valuation_theory|finite places]] (the classical ideal theoretic interpretation) but also at the infinite places of ''K''. That is, every [[real embedding]] of ''K'' extends to a real embedding of ''E'' (rather than to a complex embedding of ''E'').
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| ==Examples==
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| If the ring of integers of ''K'' is a [[unique factorization domain]], in particular, if <math> K = \mathbb{Q} </math> then ''K'' is its own Hilbert class field.
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| By contrast, let <math> K = \mathbb{Q}(\sqrt{-15}) </math>. By analyzing ramification degrees over <math>\mathbb{Q}</math>, one can show that <math> L = \mathbb{Q}(\sqrt{-3}, \sqrt{5}) </math> is an everywhere unramified extension of ''K'', and it is certainly abelian. Hence the Hilbert class field of ''K'' is a nontrivial extension and the ring of integers of ''K'' cannot be a unique factorization domain. (In fact, using the [[Minkowski bound]], one can show that ''K'' has class number exactly 2.) Hence, the Hilbert class field is <math> L </math>.
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| To see why ramification at the archimedean primes must be taken into account, consider the [[real number|real]] [[quadratic field]] ''K'' obtained by adjoining the square root of 3 to '''Q'''. This field has class number 1, but the extension ''K''(''i'')/''K'' is unramified at all prime ideals in ''K'', so ''K'' admits finite abelian extensions of degree greater than 1
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| in which all primes of ''K'' are unramified. This doesn't contradict the Hilbert class field of ''K'' being ''K'' itself: every proper finite abelian extension of ''K''
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| must ramify at some place, and in the extension ''K''(''i'')/''K'' there is ramification at the archimedean places:
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| the real embeddings of ''K'' extend to complex (rather than real) embeddings of ''K''(''i''). | |
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| ==History==
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| The existence of a Hilbert class field for a given number field ''K'' was conjectured by [[David Hilbert]]{{Citation needed|date=August 2009}} and proved by [[Philipp Furtwängler]].<ref>{{harvnb|Furtwängler|1906}}</ref> The existence of the Hilbert class field is a valuable tool in studying the structure of the [[ideal class group]] of a given field.
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| ==Additional properties==
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| The Hilbert class field ''E'' also satisfies the following:
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| *''E'' is a finite Galois [[field extension|extension]] of ''K'' and [''E'' :'' K'']=''h''<sub>''K''</sub>, where ''h''<sub>''K''</sub> is the [[ideal class|class number]] of ''K''.
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| *The [[ideal class group]] of ''K'' is [[automorphism|isomorphic]] to the [[Galois group]] of ''E'' over ''K''.
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| *Every [[ideal (ring theory)|ideal]] of ''O''<sub>''K''</sub> is a [[principal ideal]] of the [[ring (mathematics)|ring]] extension ''O''<sub>''E''</sub> ([[principal ideal theorem]]).
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| *Every [[prime ideal]] ''P'' of ''O''<sub>''K''</sub> decomposes into the product of ''h''<sub>''K''</sub>/''f'' prime ideals in ''O''<sub>''E''</sub>, where ''f'' is the [[order of a group element|order]] of [''P''] in the ideal class group of ''O''<sub>''K''</sub>.
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| In fact, ''E'' is the unique [[field (mathematics)|field]] satisfying the first, second, and fourth properties.
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| ==Explicit constructions==
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| If ''K'' is imaginary quadratic and ''A'' is an [[elliptic curve]] with [[complex multiplication]] by the [[ring of integers]] of ''K'', then adjoining the [[j-invariant]] of ''A'' to ''K'' gives the Hilbert class field.<ref>Theorem II.4.1 of {{harvnb|Silverman|1994}}</ref>
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| ==Generalizations==
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| In [[class field theory]], one studies the [[ray class field]] with respect to a given [[Modulus_(algebraic_number_theory)|modulus]], which is a formal product of prime ideals (including, possibly, archimedean ones). The ray class field is the maximal abelian extension unramified outside the primes dividing the modulus and satisfying a particular ramification condition at the primes dividing the modulus. The Hilbert class field is then the ray class field with respect to the trivial modulus ''1''.
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| The ''narrow class field'' is the ray class field with respect to the modulus consisting of all infinite primes. For example, the argument above shows that <math> \mathbb{Q}(\sqrt{3}, i) </math> is the narrow class field of <math> \mathbb{Q}(\sqrt{3}) </math>.
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| ==Notes==
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| {{reflist}}
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| ==References==
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| * {{Citation
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| | last=Childress
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| | first=Nancy
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| | title=Class field theory
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| | year=2009
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| | isbn=978-0-387-72489-8
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| | doi=10.1007/978-0-387-72490-4
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| | publisher=[[Springer Science+Business Media|Springer]]
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| | location=New York
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| | mr=2462595
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| }}
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| * {{Citation
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| | last=Furtwängler
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| | first=Philipp
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| | author-link=Philipp Furtwängler
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| | title=Allgemeiner Existenzbeweis für den Klassenkörper eines beliebigen algebraischen Zahlkörpers
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| | url=http://gdz.sub.uni-goettingen.de/dms/load/toc/?PPN=PPN235181684_0063&DMDID=dmdlog7
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| | year=1906
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| | journal=Mathematische Annalen
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| | volume=63
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| | issue=1
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| | pages=1–37
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| | doi=10.1007/BF01448421
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| | accessdate=2009-08-21
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| | mr=1511392
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| | jfm=37.0243.02
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| }}
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| * J. S. Milne, Class Field Theory (Course notes available at http://www.jmilne.org/math/). See the Introduction chapter of the notes, especially p. 4.
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| *{{Citation
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| | last=Silverman
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| | first=Joseph H.
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| | author-link=Joseph H. Silverman
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| | title=Advanced topics in the arithmetic of elliptic curves
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| | year=1994
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| | publisher=[[Springer-Verlag]]
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| | location=New York
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| | isbn=978-0-387-94325-1
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| | series=[[Graduate Texts in Mathematics]]
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| | volume=151
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| }}
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| * {{Citation
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| | last=Gras
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| | first=Georges
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| | title=Class field theory: From theory to practice
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| | year=second corrected printing 2005
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| | publisher=Springer
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| | location=New York}}
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| {{PlanetMath attribution|id=2870|title=Existence of Hilbert class field}}
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| {{DEFAULTSORT:Hilbert Class Field}}
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| [[Category:Class field theory]]
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