# Field

This article is about a basic definition in commutative algebra. View a complete list of basic definitions in commutative algebra

This article defines a property of integral domains, viz., a property that, given any integral domain, is either true or false for that.

View other properties of integral domains | View all properties of commutative unital ringsVIEW RELATED: Commutative unital ring property implications | Commutative unital ring property non-implications |Commutative unital ring metaproperty satisfactions | Commutative unital ring metaproperty dissatisfactions | Commutative unital ring property satisfactions | Commutative unital ring property dissatisfactions

## Definition

### Symbol-free definition

A **field** is a commutative unital ring with the additional property that its multiplicative group comprises all the nonzero elements, that is, with the property that all nonzero elements are invertible.

Alternatively, a field is a commutative unital ring with no proper nontrivial ideal.

### Definition with symbols

A **field** is a set endowed with constants and (not equal), a unary operation and binary operations and such that:

- for all in
- for all in
- for all in
- for all in
- for all in
- for all in
- for all in
- for all in
- For all nonzero in , there exists a in such that

## Metaproperties

### Closure under taking subrings

This property of commutative unital rings isnotclosed under taking subrings; in other words, a subring of a commutative unital ring with this property need not have this property

A subring of a field need not be a field. It could be any integral domain. For instance is a subring of .

## Effect of property operators

### The subring-closure

*Applying the subring-closure to this property gives*: integral domain

## Module theory

Modules over fields are precisely the same as vector spaces. In particular any finitely generated module over a field is a free module and the number of generators is independent of the choice of generating set.