Galois groups of finite abelian extensions of


Let G be a finite abelian group with |G|>1. Then there exist infinitely many number fieldsMathworldPlanetmath K with K/Q Galois and Gal(K/Q)G.


This will first be proven for G cyclic.

Let |G|=n. By Dirichlet’s theorem on primes in arithmetic progressions, there exists a prime p with p1modn. Let ζp denote a pth root of unityMathworldPlanetmath. Let L=(ζp). Then L/ is Galois with Gal(L/) cyclic of order ( p-1. Since n divides p-1, there exists a subgroupMathworldPlanetmathPlanetmath H of Gal(L/) such that |H|=p-1n. Since Gal(L/) is cyclic, it is abelianMathworldPlanetmath, and H is a normal subgroupMathworldPlanetmath of Gal(L/). Let K=LH, the subfieldMathworldPlanetmath of L fixed ( by H. Then K/ is Galois with Gal(K/) cyclic of order n. Thus, Gal(K/)G.

Let p and q be distinct primes with p1modn and q1modn. Then there exist subfields K1 and K2 of (ζp) and (ζq), respectively, such that Gal(K1/)G and Gal(K2/)G. Note that K1K2= since K1K2(ζp)(ζq)=. Thus, K1K2. Therefore, for every prime p with p1modn, there exists a distinct number field K such that K/ is Galois and Gal(K/)G. The theorem in the cyclic case follows from using the full of Dirichlet’s theorem on primes in arithmetic progressions: There exist infinitely many primes p with p1modn.

The general case follows immediately from the above , the fundamental theorem of finite abelian groups (, and a theorem regarding the Galois groupMathworldPlanetmath of the compositum of two Galois extensionsMathworldPlanetmath. ∎

Title Galois groups of finite abelian extensionsMathworldPlanetmathPlanetmath of
Canonical name GaloisGroupsOfFiniteAbelianExtensionsOfmathbbQ
Date of creation 2013-03-22 16:18:40
Last modified on 2013-03-22 16:18:40
Owner Wkbj79 (1863)
Last modified by Wkbj79 (1863)
Numerical id 11
Author Wkbj79 (1863)
Entry type Theorem
Classification msc 11R32
Classification msc 11N13
Classification msc 11R20
Classification msc 12F10
Related topic AbelianNumberField