-extension
Definition 1.
Let be a prime number. A Galois extension of fields , with , is said to be a -extension if is a -group.
Example 1.
Let be a square-free integer. Then the field extension is a -extension.
Example 2.
Let be a prime and, for any , let be a primitive th root of unity. The cyclotomic extension:
is a -extension. Indeed:
Thus, and , where is the Euler phi function. Therefore the extension above is of degree .
Title | -extension |
---|---|
Canonical name | Pextension |
Date of creation | 2013-03-22 15:02:56 |
Last modified on | 2013-03-22 15:02:56 |
Owner | alozano (2414) |
Last modified by | alozano (2414) |
Numerical id | 5 |
Author | alozano (2414) |
Entry type | Definition |
Classification | msc 12F05 |
Synonym | p-extension |
Related topic | PGroup4 |
Related topic | UnramifiedExtensionsAndClassNumberDivisibility |
Related topic | PushDownTheoremOnClassNumbers |
Related topic | ClassNumberDivisibilityInPExtensions |
Related topic | QuadraticExtension |