Fields

In mathematics, a field is an algebraic structure where addition and multiplication has the properties of commutativity, associativity, distributivity, and contains at least two distinct elements 0 and 1. There must also be a multiplicative identity (multiply to get the same number) and a multiplicative inverse (multiply to yield the multiplicative identity).

For three elements in a field, $x,y,z\in \mathbb{F}$, the property of commutativity suggests:

$$x\cdot y=y\cdot x$$

Associativity suggests:

$$x\cdot (y\cdot z)=(x\cdot y)\cdot z$$

Distributivity suggests:

$$x\cdot (y+z)=x\cdot y+x\cdot z$$

For example, the set of integers is not a field, but the set of rational, real, and complex numbers are fields.

See also

• Not to be confused with vector fields, which are sometimes just called fields