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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''.

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'').

==Examples==
*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.
*Let <math> K = \mathbb{Q}(\sqrt{-15}) </math> of [[Discriminant of an algebraic number field|discriminant]] <math>-15</math>. The field <math> L =  \mathbb{Q}(\sqrt{-3}, \sqrt{5}) </math> has discriminant <math>225=(-15)^2</math> and so is an everywhere unramified extension of ''K'', and it is abelian. Using the [[Minkowski bound]], one can show that ''K'' has class number 2. Hence, its Hilbert class field is <math> L </math>. A non-principal ideal of ''K'' is (2,(1+{{radic|−15}})/2), and in ''L'' this becomes the principal ideal ((1+{{radic|5}})/2).
*The field <math> \mathbb{Q}(\sqrt{-23})</math> has class number 3. Its Hilbert class field can be formed by adjoining a root of x<sup>3</sup> - x - 1, which has discriminant -23.
*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 and discriminant 12, but the extension ''K''(''i'')/''K'' of discriminant 9=3<sup>2</sup> is unramified at all prime ideals in ''K'', so ''K'' admits finite abelian extensions of degree greater than 1 in which all finite 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'' must ramify at some place, and in the extension ''K''(''i'')/''K'' there is ramification at the archimedean places: the real embeddings of ''K'' extend to complex (rather than real) embeddings of ''K''(''i'').
*By the theory of [[complex multiplication]], the Hilbert class field of an [[imaginary quadratic field]] is generated by the value of the [[elliptic modular function]] at a generator for the ring of integers (as a '''Z'''-module).

==History==

The existence of a (narrow) Hilbert class field for a given number field ''K'' was conjectured by {{harvs|txt|first=David|last= Hilbert|year=1902|authorlink=David Hilbert}} 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.

==Additional properties==

The Hilbert class field ''E'' also satisfies the following:
*''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''.
*The [[ideal class group]] of ''K'' is [[automorphism|isomorphic]] to the [[Galois group]] of ''E'' over ''K''.
*Every [[ideal (ring theory)|ideal]] of ''O''<sub>''K''</sub> extends to a [[principal ideal]] of the ring extension ''O''<sub>''E''</sub> ([[principal ideal theorem]]).
*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>.

In fact, ''E'' is the unique [[field (mathematics)|field]] satisfying the first, second, and fourth properties.

==Explicit constructions==
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>

==Generalizations==
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''.

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>.

==Notes==
{{reflist}}

==References==
* {{Citation
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| title=Class field theory
| year=2009
| isbn=978-0-387-72489-8
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| publisher=[[Springer Science+Business Media|Springer]]
| location=New York
| mr=2462595
}}
* {{Citation
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| first=Philipp
| author-link=Philipp Furtwängler
| title=Allgemeiner Existenzbeweis für den Klassenkörper eines beliebigen algebraischen Zahlkörpers
| url=http://gdz.sub.uni-goettingen.de/dms/load/toc/?PPN=PPN235181684_0063&DMDID=dmdlog7
| year=1906
| journal=Mathematische Annalen
| volume=63
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| doi=10.1007/BF01448421

| accessdate=2009-08-21
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*{{citation|journal=Acta Mathematica
|year=1902|orig-date=1898|volume =26|issue =1|pages= 99–131
|title=Über die Theorie der relativ-Abel'schen Zahlkörper
|first=David |last=Hilbert|doi=10.1007/BF02415486|doi-access=free}}
* J. S. Milne, Class Field Theory (Course notes available at http://www.jmilne.org/math/).  See the Introduction chapter of the notes, especially p.&nbsp;4.
*{{Citation
| last=Silverman
| first=Joseph H.
| author-link=Joseph H. Silverman
| title=Advanced topics in the arithmetic of elliptic curves
| year=1994
| publisher=[[Springer-Verlag]]
| location=New York
| isbn=978-0-387-94325-1
| series=[[Graduate Texts in Mathematics]]
| volume=151
}}
* {{Citation
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{{PlanetMath attribution|id=2870|title=Existence of Hilbert class field}}

[[Category:Class field theory]]