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| Title of object: |
Abelian extension |
| Canonical Name: |
AbelianExtension |
| Type: |
Definition |
| Created on: |
2002-11-15 02:15:48 |
| Modified on: |
2002-11-15 02:15:48 |
| Classification: |
msc:12F10 |
Revision comment (for changes between this and next version):
Changes for correction #7720 ('bad linking of title').
Also fixed link from "primitive nth root of unity". |
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Content:
Let $K$ be a Galois extension of $F$. 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. |
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