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# $p$-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 $p^{{(n-1)}}$.

Keywords:

field extension

Related:

PGroup4, UnramifiedExtensionsAndClassNumberDivisibility, PushDownTheoremOnClassNumbers, ClassNumberDivisibilityInPExtensions, QuadraticExtension

Synonym:

p-extension

Type of Math Object:

Definition

Major Section:

Reference

## Mathematics Subject Classification

12F05*no label found*

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new question: Prime numbers out of sequence by Rubens373

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new question: how to contest an entry? by zorba

new question: simple question by parag

new question: Prime numbers out of sequence by Rubens373

Oct 7

new question: Lorenz system by David Bankom

Oct 19

new correction: examples and OEIS sequences by fizzie

Oct 13

new correction: Define Galois correspondence by porton

Oct 7

new correction: Closure properties on languages: DCFL not closed under reversal by babou

new correction: DCFLs are not closed under reversal by petey

Oct 2

new correction: Many corrections by Smarandache

Sep 28

new question: how to contest an entry? by zorba

new question: simple question by parag