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# abelian extension

Let $K$ be a Galois extension of $F$. The extension is said to be an abelian extension if the Galois group Gal$(K/F)$ is abelian.

Examples: $\mathbb{Q}(\sqrt{2})/\mathbb{Q}$ has Galois group $\mathbb{Z}/2\mathbb{Z}$ so $\mathbb{Q}(\sqrt{2})/\mathbb{Q}$ is an abelian extension.

Let $\zeta_{n}$ be a primitive nth root of unity. Then $\mathbb{Q}(\zeta_{n})/\mathbb{Q}$ has Galois group $(\mathbb{Z}/n\mathbb{Z})^{*}$ (the group of units of $\mathbb{Z}/n\mathbb{Z}$) so $\mathbb{Q}(\zeta_{n})/\mathbb{Q}$ is abelian.

Related:

KroneckerWeberTheorem, KummerTheory

Type of Math Object:

Definition

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Reference

## Mathematics Subject Classification

12F10*no label found*

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