Galois groups of finite abelian extensions of $\mathbb{Q}$

Theorem.

Let $G$ be a finite abelian group with $\mathrm{|}G\mathrm{|}\mathrm{>}\mathrm{1}$. Then there exist infinitely many number fields $K$ with $K\mathrm{/}\mathrm{Q}$ Galois and $\mathrm{Gal}\mathit{}\mathrm{(}K\mathrm{/}\mathrm{Q}\mathrm{)}\mathrm{\cong }G$.

Proof.

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 $p\equiv 1\mathrm{mod}n$. Let ${\zeta }_p$ denote a $p^{\text{th}}$ root of unity. Let $L=\mathbb{Q}({\zeta }_p)$. Then $L/\mathbb{Q}$ is Galois with $\mathrm{Gal}(L/\mathbb{Q})$ cyclic of order (http://planetmath.org/OrderGroup) $p-1$. Since $n$ divides $p-1$, there exists a subgroup $H$ of $\mathrm{Gal}(L/\mathbb{Q})$ such that $|H|={\displaystyle \frac{p-1}{n}}$. Since $\mathrm{Gal}(L/\mathbb{Q})$ is cyclic, it is abelian, and $H$ is a normal subgroup of $\mathrm{Gal}(L/\mathbb{Q})$. Let $K=L^H$, the subfield of $L$ fixed (http://planetmath.org/FixedField) by $H$. Then $K/\mathbb{Q}$ is Galois with $\mathrm{Gal}(K/\mathbb{Q})$ cyclic of order $n$. Thus, $\mathrm{Gal}(K/\mathbb{Q})\cong G$.

Let $p$ and $q$ be distinct primes with $p\equiv 1\mathrm{mod}n$ and $q\equiv 1\mathrm{mod}n$. Then there exist subfields $K_1$ and $K_2$ of $\mathbb{Q}({\zeta }_p)$ and $\mathbb{Q}({\zeta }_q)$, respectively, such that $\mathrm{Gal}(K_1/\mathbb{Q})\cong G$ and $\mathrm{Gal}(K_2/\mathbb{Q})\cong G$. Note that $K_1\cap K_2=\mathbb{Q}$ since $\mathbb{Q}\subseteq K_1\cap K_2\subseteq \mathbb{Q}({\zeta }_p)\cap \mathbb{Q}({\zeta }_q)=\mathbb{Q}$. Thus, $K_1\ne K_2$. Therefore, for every prime $p$ with $p\equiv 1\mathrm{mod}n$, there exists a distinct number field $K$ such that $K/\mathbb{Q}$ is Galois and $\mathrm{Gal}(K/\mathbb{Q})\cong G$. 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 $p\equiv 1\mathrm{mod}n$.

The general case follows immediately from the above , the fundamental theorem of finite abelian groups (http://planetmath.org/FundamentalTheoremOfFinitelyGeneratedAbelianGroups), and a theorem regarding the Galois group of the compositum of two Galois extensions. ∎

Title	Galois groups of finite abelian extensions of $\mathbb{Q}$
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