PlanetMath: abelian extension

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abelian extension

(Definition)

Let be a Galois extension of . The extension is said to be an abelian extension if the Galois group $\textrm{Gal$(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.

"abelian extension" is owned by scanez. [ full author list (2) | owner history (2) ]

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See Also: Kronecker-Weber theorem, Kummer theory

Attachments:: Kronecker-Weber theorem (Theorem) by alozano

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Cross-references: group of units, abelian, Galois group, extension, Galois extension
There are 6 references to this entry.

This is version 2 of abelian extension, born on 2002-11-15, modified 2006-04-30.
Object id is 3599, canonical name is AbelianExtension.
Accessed 3145 times total.

Classification:

AMS MSC:

12F10 (Field theory and polynomials :: Field extensions :: Separable extensions, Galois theory)

Pending Errata and Addenda

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