Hilbert class field
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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.
Note that 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). As an example of why this is necessary, consider the real quadratic field K obtained by adjoining the square root of 3 to Q. This field has class number 1, but the extension K(i)/K is unramified at all prime ideals in K, so K admits finite abelian extensions of degree greater than 1 in which all 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 extenson 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).
The existence of unique Hilbert class field for given number field K 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 given field.
[edit] Additional properties
Furthermore, E 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.
This article incorporates material from Existence of Hilbert class field on PlanetMath, which is licensed under the GFDL.
