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).
- If the ring of integers of K is a unique factorization domain, in particular, if then K is its own Hilbert class field.
- Let of discriminant 15. The field 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 . A non-principla 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 Kmust 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).
The existence of a Hilbert class field for a given number field K was conjectured by Template:Harvs and proved by Philipp Furtwängler. The existence of the Hilbert class field is a valuable tool in studying the structure of the ideal class group of a given field.
The Hilbert class field E also satisfies the following:
- E is a finite Galois extension of K and [E : K]=hK, where hK 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 OK is a principal ideal of the ring extension OE (principal ideal theorem).
- Every prime ideal P of OK decomposes into the product of hK/f prime ideals in OE, where f is the order of [P] in the ideal class group of OK.
In fact, E is the unique field satisfying the first, second, and fourth properties.
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.
- 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.