Hilbert class field

In algebraic number theory, the 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=15² 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+√−15)/2), and in L this becomes the principal ideal ((1+√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 David Hilbert (1902) and proved by Philipp Furtwängler.^[1] 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 is 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.

In fact, 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.^[2]

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

↑ Furtwängler 1906
↑ Theorem II.4.1 of Silverman 1994

References

Childress, Nancy (2009), Class field theory, New York: Springer, doi:10.1007/978-0-387-72490-4, ISBN 978-0-387-72489-8, MR 2462595
Furtwängler, Philipp (1906), "Allgemeiner Existenzbeweis für den Klassenkörper eines beliebigen algebraischen Zahlkörpers", Mathematische Annalen, 63 (1): 1–37, doi:10.1007/BF01448421, JFM 37.0243.02, MR 1511392, retrieved 2009-08-21
Hilbert, David (1902) [1898], "Über die Theorie der relativ-Abel'schen Zahlkörper", Acta Mathematica, 26 (1): 99–131, doi:10.1007/BF02415486
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.
Silverman, Joseph H. (1994), Advanced topics in the arithmetic of elliptic curves, Graduate Texts in Mathematics, 151, New York: Springer-Verlag, ISBN 978-0-387-94325-1
Gras, Georges (2005), Class field theory: From theory to practice, New York: Springer

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