Hilbert class field

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".

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.

Additional properties

Furthermore, "E" satisfies the following:
*"E" is a finite Galois extension of "K" and ["E" :" K"] ="h""K", where "h""K" 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 "O""K" is a principal ideal of the ring extension "O""E" (principal ideal theorem).
*Every prime ideal "P" of "O""K" decomposes into the product of "h""K"/"f" prime ideals in "O""E", where "f" is the order of ["P"] in the ideal class group of "O""K".

In fact, "E" is the unique field satisfying the first, second, and fourth properties.

References

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