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nLab

separable field extension

# Contents

## Definition

A polynomial $P$ over a field $K$ is **separable** if all its irreducible factors have distinct roots over the algebraic closure $\bar{K}$ of $K$.

An extension $K\subset L$ of fields is **separable** if every element $x\in L$ is a root of a separable polynomial over $K$.

## Properties

Every finite separable field extension is an étale morphism of rings.

If $K\subset L\subset M$ are fields and $K\subset M$ is separable, then $L\subset M$ is also separable.

Revised on December 9, 2013 05:55:15
by

Urs Schreiber
(89.204.139.250)