Separably closed field

This article defines a field property: a property that can be evaluated to true/false for any field.
View a complete list of field properties|View a complete list of field extension properties

Definition

A field is termed separably closed if it satisfies the following equivalent conditions:

1. Every irreducible polynomial over it that is separable is linear.
2. It has no proper algebraic extension that is separable.

For any field, we can define the separable closure of the field, which is the unique smallest separably closed field containing it.

Relation with other properties

Stronger properties

• Algebraically closed field

Opposite properties

• Perfect field: In fact, a field is both perfect and separably closed if and only if it is algebraically closed.