# $p$-extension

###### Definition 1.

Let $p$ be a prime number. A Galois extension of fields $E/F$, with $G=\operatorname{Gal}(E/F)$, is said to be a $p$-extension if $G$ is a $p$-group.

###### Example 1.

Let $d$ be a square-free integer. Then the field extension $\mathbb{Q}(\sqrt{d})/\mathbb{Q}$ is a $2$-extension.

###### Example 2.

Let $p>2$ be a prime and, for any $n$, let $\zeta_{p^{n}}$ be a primitive $p^{n}$th root of unity. The cyclotomic extension:

 $\mathbb{Q}(\zeta_{p^{n}})/\mathbb{Q}(\zeta_{p})$

is a $p$-extension. Indeed:

 $G_{n}=\operatorname{Gal}(\mathbb{Q}(\zeta_{p^{n}})/\mathbb{Q})\cong(\mathbb{Z}% /p^{n}\mathbb{Z})^{\times}$

Thus, $|G_{n}|=\varphi(p^{n})=p^{(n-1)}(p-1)$ and $|G_{1}|=\varphi(p)=p-1$, where $\varphi$ is the Euler phi function. Therefore the extension above is of degree $p^{(n-1)}$.

Title $p$-extension Pextension 2013-03-22 15:02:56 2013-03-22 15:02:56 alozano (2414) alozano (2414) 5 alozano (2414) Definition msc 12F05 p-extension PGroup4 UnramifiedExtensionsAndClassNumberDivisibility PushDownTheoremOnClassNumbers ClassNumberDivisibilityInPExtensions QuadraticExtension