 ##  [Field](/field) 

  ##  [Field](https://natural.quantumdictionary.io/field-0) 

  
 [![Natural & Formal Sciences Dictionary](/sites/default/files/styles/large/public/2026-01/Natural%20%26%20Formal%20Sciences.png.webp?itok=2kCDRVQv)](/topic-specific-dictionaries/natural-formal-sciences)


**Natural &amp; Formal Sciences Dictionary**

 
Definition

A commutative ring with unity in which every nonzero element has a multiplicative inverse; it supports addition, subtraction, multiplication and division (by nonzero elements) and provides the canonical scalar domains for vector spaces and many algebraic constructions.

 
 ##  [Field](https://mathlogic.quantumdictionary.io/field-1) 

  
 [![Mathematics & Logic Dictionary](/sites/default/files/styles/large/public/2026-01/Mathematics%20%26%20Logic.png.webp?itok=UhtTRPnp)](/topic-specific-dictionaries/natural-formal-sciences/mathematics-logic)

- Natural &amp; Formal Sciences -

**Mathematics &amp; Logic Dictionary**

 
Definition

A commutative ring with unity in which every nonzero element has a multiplicative inverse; equivalently, a set with two operations forming an abelian additive group and a multiplicative abelian group on nonzero elements.

 
 ##  [Field](https://puremath.quantumdictionary.io/field-2) 

  
 [![Pure Mathematics Dictionary](/sites/default/files/styles/large/public/2026-01/Pure%20Mathematics.png.webp?itok=5pZnFQ59)](/topic-specific-dictionaries/mathematics-logic/pure-mathematics)

- Mathematics &amp; Logic -

**Pure Mathematics Dictionary**

 
Definition

A commutative ring with unity in which every nonzero element has a multiplicative inverse; equivalently, a commutative ring whose nonzero elements form an abelian group under multiplication.

 
 ##  [Field](https://algebra.quantumdictionary.io/field-3) 

  
 [![Algebra](/sites/default/files/styles/large/public/2026-01/Algebra.png.webp?itok=3pHxBnUF)](/topic-specific-dictionaries/pure-mathematics/algebra)

- Pure Mathematics -

**Algebra Dictionary**

 
Definition

A commutative division ring: an associative ring with unity in which multiplication is commutative and every nonzero element has a multiplicative inverse.