Algebraically closed field: Difference between revisions
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* Every nonconstant polynomial with coefficients in the field has a root in the field. | * Every nonconstant polynomial with coefficients in the field has a root in the field. | ||
* Every monic polynomial with coefficients in the field can be expressed as a product of linear polynomials with coefficients in the field. | * Every monic polynomial with coefficients in the field can be expressed as a product of linear polynomials with coefficients in the field. | ||
==Relation with other properties== | |||
===Weaker properties=== | |||
* [[Stronger than::Separably closed field]] | |||
* [[Stronger than::Perfect field]] | |||
* [[Stronger than::Quadratically closed field]] | |||
* [[Stronger than::Pythagorean field]] | |||
* [[Stronger than::Field with trivial Brauer group]] | |||
==Metaproperties== | |||
{{subfield-realizable}} | |||
Every field can be realized as a subfield of an algebraically closed field, namely, its [[algebraic closure]]. | |||
{{finite-extension-closed}} | |||
In fact, an algebraically closed field has no proper finite extensions, so a finite extension of an algebraically closed field is algebraically closed. | |||
Latest revision as of 22:23, 14 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
Symbol-free definition
A field is termed algebraically closed if it satisfies the following conditions:
- Every nonconstant polynomial with coefficients in the field has a root in the field.
- Every monic polynomial with coefficients in the field can be expressed as a product of linear polynomials with coefficients in the field.
Relation with other properties
Weaker properties
- Separably closed field
- Perfect field
- Quadratically closed field
- Pythagorean field
- Field with trivial Brauer group
Metaproperties
Subfield-realizability
This field property is subfield-realizable: every field can be realized as a subfield of a field with this property.
View other subfield-realizable field properties
Every field can be realized as a subfield of an algebraically closed field, namely, its algebraic closure.
Finite-extension-closedness
This field property is closed under taking finite extensions. In other words, any finite extension of a field with this property also has this property.
View other such properties
In fact, an algebraically closed field has no proper finite extensions, so a finite extension of an algebraically closed field is algebraically closed.