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Concept
Hilbert class field
Summary
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 . The field has discriminant 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-principal ideal of K is (2,(1+)/2), and in L this becomes the principal ideal ((1+)/2).
The field has class number 3. Its Hilbert class field can be formed by adjoining a root of x3 - x - 1, which has discriminant -23.
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 12, 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).
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