Algebraically 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

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.