$p$-extension

Definition 1.

Let $p$ be a prime number. A Galois extension of fields $E\mathrm{/}F$, with $G\mathrm{=}\mathrm{Gal}\mathit{}\mathrm{(}E\mathrm{/}F\mathrm{)}$, 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 $\mathrm{Q}\mathit{}\mathrm{(}\sqrt{d}\mathrm{)}\mathrm{/}\mathrm{Q}$ is a $\mathrm{2}$-extension.

Example 2.

Let $p\mathrm{>}\mathrm{2}$ 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:

$$G_n=\mathrm{Gal}(\mathbb{Q}({\zeta }_{p^n})/\mathbb{Q})\cong {(\mathbb{Z}/p^n\mathbb{Z})}^{\times }$$

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