characterization of abelian extensions of exponent n
Theorem 1.
Let $K$ be a field containing the ${n}^{\mathrm{th}}$ roots of unity^{}, with characteristic^{} not dividing $n$. Let $L$ be a finite extension^{} of $K$. Then the following are equivalent^{}:

1.
$L/K$ is Galois with abelian^{} Galois group^{} of exponent dividing $n$

2.
$L=K(\sqrt[n]{{a}_{1}},\mathrm{\dots},\sqrt[n]{{a}_{k}})$ for some ${a}_{i}\in K$.
Proof.
Let $\zeta \in K$ be a primitive ${n}^{\mathrm{th}}$ root of unity.
$(2\Rightarrow 1):$ Choose ${\alpha}_{i}\in L$ such that ${\alpha}_{i}^{n}={a}_{i}\in K$. Then for each $i$, the elements ${\alpha}_{i},\zeta {\alpha}_{i},\mathrm{\dots},{\zeta}^{n1}{\alpha}_{i}$ are distinct and are all the roots of ${x}^{n}{a}_{i}$ in $L$. Thus ${x}^{n}{a}_{i}$ is separable^{} over $K$ and splits in $L$, so that $L$ is the splitting field^{} of the set of polynomials^{} $\{{x}^{n}{a}_{i}\mid \mathrm{\hspace{0.25em}1}\le i\le k\}$. Thus $L/K$ is Galois. Given $\sigma \in \mathrm{Gal}(L/K)$, for each $i$ we have $\sigma ({\alpha}_{i})={\zeta}^{j}{\alpha}_{i}$ for some $1\le j\le n$, so that ${\sigma}^{k}({\alpha}_{i})={\zeta}^{kj}{\alpha}_{i}$. It follows that ${\sigma}^{n}$ is the identity^{} for every $\sigma \in \mathrm{Gal}(L/K)$, so that the exponent of $\mathrm{Gal}(L/K)$ divides $n$. It remains to show that $\mathrm{Gal}(L/K)$ is abelian; this follows trivially from the simple definition of the Galois action as multiplication^{} by some ${n}^{\mathrm{th}}$ root of unity: if $\sigma ,\tau \in \mathrm{Gal}(L/K)$ with $\sigma ({\alpha}_{i})={\zeta}^{r}{\alpha}_{i},\tau ({\alpha}_{i})={\zeta}^{s}{\alpha}_{i}$, then
$(\sigma \tau )({\alpha}_{i})$  $=\sigma ({\zeta}^{s}{\alpha}_{i})={\zeta}^{s}{\zeta}^{r}{\alpha}_{i}$  
$(\tau \sigma )({\alpha}_{i})$  $=\tau ({\zeta}^{r}{\alpha}_{i})={\zeta}^{r}{\zeta}^{s}{\alpha}_{i}$ 
Thus $\sigma \tau =\tau \sigma $ on each ${\alpha}_{i}$. But the ${\alpha}_{i}$ generate $L/K$, so $\sigma \tau =\tau \sigma $ on $L$ and $\mathrm{Gal}(L/K)$ is abelian.
$(1\Rightarrow 2):$ Let $G=\mathrm{Gal}(L/K)$, and write $G={C}_{1}\times \mathrm{\dots}\times {C}_{r}$ where each ${C}_{i}$ is cyclic; $\left{C}_{i}\right={m}_{i}\mid n$ for each $i$. For each $i$, define a subgroup^{} ${H}_{i}\le G$ by
$${H}_{i}={C}_{1}\times \mathrm{\dots}\times {C}_{i1}\times {C}_{i+1}\times \mathrm{\dots}\times {C}_{r}$$ 
Then $G/{H}_{i}\cong {C}_{i}$. Let ${L}_{i}$ be the fixed field of ${H}_{i}$. ${L}_{i}$ is normal over $K$ since ${H}_{i}$ is normal in $G$, and $\mathrm{Gal}({L}_{i}/K)\cong G/{H}_{i}\cong {C}_{i}$ and thus ${L}_{i}/K$ is cyclic Galois of order ${m}_{i}$. $K$ contains the primitive ${m}_{i}^{\mathrm{th}}$ root of unity ${\zeta}^{n/{m}_{i}}$ and thus ${L}_{i}=K({\alpha}_{i})$ for some ${\alpha}_{i}\in L$ with ${\alpha}_{i}^{{m}_{i}}\in K$ (by Kummer theory). But then also ${\alpha}_{i}^{n}\in K$. Then
$$\mathrm{Gal}(L:K({\alpha}_{1},\mathrm{\dots},{\alpha}_{r}))={H}_{1}\cap \mathrm{\dots}\cap {H}_{r}=\{1\}$$ 
since any element of the lefthand group fixes each ${\alpha}_{i}$ and thus fixes ${L}_{i}$ so is the identity in $G/{H}_{i}$. Thus $L=K({\alpha}_{1},\mathrm{\dots},{\alpha}_{r})$. ∎
Corollary 2.
If $L\mathrm{/}K$ is the maximal abelian extension^{} of $K$ of exponent $n$, where $n$ is prime to the characteristic of $K$, then $L\mathrm{=}K\mathit{}\mathrm{(}\mathrm{\{}\sqrt[n]{a}\mathrm{\}}\mathrm{)}$ for some set of $a\mathrm{\in}K$.
Proof.
Clearly $K(\{\sqrt[n]{a}\mid a\in {K}^{*})$ is an infinite^{} abelian extension of exponent $n$. If $L$ is the maximal such extension^{}, choose $b\in L$. Then $K(b)$ is a finite extension of exponent dividing $n$ and thus $K(b)$ is of the required form. Thus $L={\cup}_{b\in L}K(b)$ is also of the required form; for example, if $S\subset {K}^{*}$ is a set of coset representatives for ${K}^{*}/{({K}^{*})}^{n}$, then $L=K(S)$. ∎
References
 1 Morandi, P., Field and Galois Theory^{}, Springer, 1996.
Title  characterization of abelian extensions of exponent n 

Canonical name  CharacterizationOfAbelianExtensionsOfExponentN 
Date of creation  20130322 18:42:13 
Last modified on  20130322 18:42:13 
Owner  rm50 (10146) 
Last modified by  rm50 (10146) 
Numerical id  4 
Author  rm50 (10146) 
Entry type  Theorem 
Classification  msc 12F10 