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 rings
VIEW 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 $F$ endowed with constants $0$ and $1$ (not equal), a unary operation $-$ and binary operations $+$ and $*$ such that:

• $a+(b+c)=(a+b)+c$ for all $a,b,c$ in $F$
• $a+0=a$ for all $a$ in $F$
• $a+b=b+a$ for all $a,b$ in $F$
• $a+(-a)=0$ for all $a$ in $F$
• $a*(b*c)=(a*b)*c$ for all $a,b,c$ in $F$
• $a*1=a$ for all $a$ in $F$
• $a*b=b*a$ for all $a,b$ in $F$
• $a*(b+c)=(a*b)+(a*c)$ for all $a,b,c$ in $F$
• For all nonzero $a$ in $F$, there exists a $b$ in $F$ such that $a*b=1$

Metaproperties

Closure under taking subrings

This property of commutative unital rings is not closed 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 $\mathbb{Z}$ is a subring of $\mathbb{R}$.

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.