purely inseparable

Let $F$ be a field of characteristic $p>0$ and let $\alpha$ be an element which is algebraic over $F$ . Then $\alpha$ is said to be purely inseparable over $F$ if $\alpha^{p^{n}}\in F$ for some $n\geq 0$ .

An algebraic field extension $K/F$ is purely inseparable if each element of $K$ is purely inseparable over $F$ .

Purely inseparable extensions have the following property: if $K/F$ is purely inseparable, and $A$ is an algebraic closure of $F$ which contains $K$ , then any homomorphism $K\to A$ which fixes $F$ necessarily fixes $K$ .

Let $K/F$ be an arbitrary algebraic extension. Then there is an intermediate field $E$ such that $K/E$ is purely inseparable, and $E/F$ is separable.

Example.

Let $s$ be an indeterminate, and let $K=\mathbb{F}_{3}(s)$ where $\mathbb{F}_{3}$ is the finite field with $3$ elements. Let $F=\mathbb{F}_{3}(s^{6})$ . Then $K/F$ is neither separable, nor purely inseparable. Let $E=\mathbb{F}_{3}(s^{3})$ . Then $E/F$ is separable, and $K/E$ is purely inseparable.

Title	purely inseparable
Canonical name	PurelyInseparable
Date of creation	2013-03-22 14:49:08
Last modified on	2013-03-22 14:49:08
Owner	mclase (549)
Last modified by	mclase (549)
Numerical id	6
Author	mclase (549)
Entry type	Definition
Classification	msc 12F15