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]] | |||
Revision as of 17:28, 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
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.