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.∎