In algebraic number theory, the Hilbert class field of a number field is the maximal abelian unramified and the Galois group is canonically isomorphic to the ideal class group using Frobenius elements for prime ideals.

is not just unramified at the finite places. That is, every real embedding.

Examples

• is a unique factorization domain is its own Hilbert class field.

  , of discriminant, and it is abelian. Using the Minkowski bound.

  .

  To see why ramification at the archimedean primes must be taken into account, consider the real quadratic field,.

  ,, but ramified at both real places.

  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-module).

History

was conjectured by David Hilbert 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.

Additional properties

also satisfies the following:

• is a finite Galois extension is the class number.

  The ideal class group is isomorphic to the Galois group.

  Every ideal extends to a principal ideal (principal ideal theorem).

  Every prime ideal is the order.

is the unique field satisfying the first, second, and fourth properties.

Explicit constructions

is an elliptic curve with complex multiplication by the ring of integers, then adjoining the j-invariant gives the Hilbert class field.

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.

Notes

References

• 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.