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.
Note that in this context, 'unramified' is meant not only for the finite places (the classical ideal theoretic interpretation) but also for the infinite places. That is, every real embedding of K extends to a real embedding of E. As an example of why this is necessary, consider some real quadratic field.
The existence of unique Hilbert class field for given number field K was conjectured by David Hilbert and proved by Phillip 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.
- 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 OE.
In fact, E is the unique field satisfying the above five properties.
This article incorporates material from Existence of Hilbert class field on PlanetMath, which is licensed under the GFDL.
