Separably closed field: Difference between revisions
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For any field, we can define the [[separable closure]] of the field, which is the unique smallest separably closed field containing it. | 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=== | |||
* [[Weaker than::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 field|algebraically closed]]. | |||
Latest revision as of 17:26, 10 May 2009
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:
- Every irreducible polynomial over it that is separable is linear.
- 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
Opposite properties
- Perfect field: In fact, a field is both perfect and separably closed if and only if it is algebraically closed.