---
tag: 7FR2PZ141W
kind: definition
name: Purely inseparable extension
msc: [12F15]
defines: 966GNVGB8A
uses: [09EYMKMSB9, 0P7CH1CND5, 0GK1WB9R98]
sources:
  - url: https://en.wikipedia.org/wiki/Purely_inseparable_extension
---

## Definition

Let $L/K$ be an [[algebraic extension|09EYMKMSB9]].
If $K$ has [[characteristic|0P7CH1CND5]] $0$, the extension $L/K$ is *purely inseparable* if $L = K$.
If $K$ has characteristic $p > 0$, the extension $L/K$ is *purely inseparable* if for every $\alpha \in L$ there is an integer $n \ge 0$ with $\alpha^{p^n} \in K$.

## Remarks

In characteristic $p > 0$, the following are equivalent for an algebraic extension $L/K$:

1. $L/K$ is purely inseparable;
2. the [[minimal polynomial|0GK1WB9R98]] over $K$ of each $\alpha \in L$ has the form $x^{p^n} - a$ with $a \in K$ and $n \ge 0$;
3. every element of $L$ that is separable over $K$ lies in $K$.

Example: let $K = \F_p(t)$ and $L = K(s)$ with $s^p = t$.
Check: every element of $L$ is $\sum_{i=0}^{p-1} c_i s^i$ with $c_i \in K$, and its $p$-th power is $\sum c_i^p t^i \in K$.

Non-example: $\F_{p^2}/\F_p$ is not purely inseparable.
For $\alpha \in \F_{p^2} \setminus \F_p$, the powers $\alpha^{p^n}$ are $\alpha$ or its conjugate $\alpha^p$, and neither lies in $\F_p$.