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 $\mathbb{Q}$ |
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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 |