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*Hilbert class field*

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Hilbert class field

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**Hilbert class field**

*E*of a number field

*K*is the 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 elements for prime ideals in

*K*.In this context, the Hilbert class field of

*K*is not just unramified at the 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 K = mathbb{Q} , then*K*is its own Hilbert class field. - Let K = mathbb{Q}(sqrt{-15}) of discriminant -15. The field L = mathbb{Q}(sqrt{-3}, sqrt{5}) has discriminant 225=-15^2 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 L . A non-principal ideal of*K*is (2,(1+{{radic|âˆ’15}})/2), and in*L*this becomes the principal ideal ((1+{{radic|5}})/2). - To see why ramification at the archimedean primes must be taken into account, consider the real quadratic field
*K*obtained by adjoining the square root of 3 to**Q**. This field has class number 1 and discriminant 3, but the extension*K*(*i*)/*K*of discriminant 9=32 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 FurtwÃ¤ngler.{{harvnb|FurtwÃ¤ngler|1906}} 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 extension of*K*and [*E*:*K*]=*h***'K****, where***h**'K*is the class number of*K*.- The ideal class group of
*K*is isomorphic to the Galois group of*E*over*K*. - Every ideal of
*O***'K****extends to a principal ideal of the ring extension***O**'E*(principal ideal theorem). - Every prime ideal
*P*of*O***'K****decomposes into the product of***h**'K*/*f*prime ideals in*O***'E****, where***f*is the order of [*P*] in the ideal class group of*O**'K*.

*E*is the unique 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.Theorem II.4.1 of {{harvnb|Silverman|1994}}

## Generalizations

In class field theory, one studies the ray class field with respect to a given 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 mathbb{Q}(sqrt{3}, i) is the narrow class field of mathbb{Q}(sqrt{3}) .

## Notes

{{reflist}}## References

- {{Citation| last=Childress| first=Nancy| title=Class field theory| year=2009| isbn=978-0-387-72489-8| doi=10.1007/978-0-387-72490-4
Springer Science+Business Media>Springer| location=New York| mr=2462595}} - {{Citation| last=FurtwÃ¤ngler| first=Philipp| author-link=Philipp FurtwÃ¤ngler| title=Allgemeiner Existenzbeweis fÃ¼r den KlassenkÃ¶rper eines beliebigen algebraischen ZahlkÃ¶rpers| url=http://gdz.sub.uni-goettingen.de/dms/load/toc/?PPN=PPN235181684_0063&DMDID=dmdlog7| year=1906| journal=Mathematische Annalen| volume=63| issue=1| pages=1â€“37| doi=10.1007/BF01448421

- {{citation|journal=Acta Mathematica

origyear=1898 issue =1|pages= 99â€“131|title=Ãœber die Theorie der relativ-Abel'schen ZahlkÃ¶rper last=Hilbert|doi=10.1007/BF02415486}} - J. S. Milne, Class Field Theory (Course notes available atweblink See the Introduction chapter of the notes, especially p. 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| last=Gras| first=Georges| title=Class field theory: From theory to practice| year=2005| publisher=Springer| location=New York}}

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