Field

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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 \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.