Definition 1.

Let p be a prime numberMathworldPlanetmath. A Galois extensionMathworldPlanetmath of fields E/F, with G=Gal(E/F), is said to be a p-extensionPlanetmathPlanetmath if G is a p-group.

Example 1.

Let d be a square-free integer. Then the field extension Q(d)/Q is a 2-extension.

Example 2.

Let p>2 be a prime and, for any n, let ζpn be a primitive pnth root of unityMathworldPlanetmath. The cyclotomic extension:


is a p-extension. Indeed:


Thus, |Gn|=φ(pn)=p(n-1)(p-1) and |G1|=φ(p)=p-1, where φ is the Euler phi function. Therefore the extension above is of degree p(n-1).

Title p-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