unramified extensions and class number divisibility

The following is a corollary of the existence of the Hilbert class fieldMathworldPlanetmath.

Corollary 1.

Let K be a number fieldMathworldPlanetmath, hK is its class numberMathworldPlanetmathPlanetmath and let p be a prime. Then K has an everywhere unramified Galois extensionMathworldPlanetmath of degree p if and only if hK is divisible by p.


Let K be a number field and let H be the Hilbert class field of K. Then:


Let p be a prime numberMathworldPlanetmath. Suppose that there exists a Galois extension F/K, such that [F:K]=p and F/K is everywhere unramified. Notice that any Galois extension of prime degree is abelianMathworldPlanetmath (because any group of prime degree p is abelian, isomorphicPlanetmathPlanetmathPlanetmath to /p). Since H is the maximal abelian unramified extensionPlanetmathPlanetmathPlanetmathPlanetmath of K the following inclusions occur:




Therefore p divides hK.

Next we prove the remaining direction. Suppose that p divides hK=|Gal(H/K)|. Since G=Gal(H/K) is an abelian group (isomorphic to the class group of K) there exists a normal subgroupMathworldPlanetmath J of G such that |G/J|=p. Let F=HJ be the fixed field by the subgroupMathworldPlanetmathPlanetmath J, which is, by the main theorem of Galois theoryMathworldPlanetmath, a Galois extension of K. This field satisfies [F:K]=p and, since F is included in H, the extension F/K is abelian and everywhere unramified, as claimed. ∎

Title unramified extensions and class number divisibility
Canonical name UnramifiedExtensionsAndClassNumberDivisibility
Date of creation 2013-03-22 15:02:59
Last modified on 2013-03-22 15:02:59
Owner alozano (2414)
Last modified by alozano (2414)
Numerical id 5
Author alozano (2414)
Entry type Corollary
Classification msc 11R37
Classification msc 11R32
Classification msc 11R29
Related topic IdealClass
Related topic PExtension
Related topic Ramify
Related topic ClassNumbersAndDiscriminantsTopicsOnClassGroups