# Field

From Groupprops

## Definition

A **field** is a set (that we'll call ) equipped with the following:

- An (infix) binary operation , called the
*addition* - An (infix) binary operation , called the
*multiplication* - Two distinct constants , called
*zero*and*one* - A unary operation denoted by the prefix symbol (called the negative, or additive inverse)
- A map

such that the following compatibility conditions hold:

- forms an abelian group with binary operation , inverse map , and identity element . This is called the additive group of .
- gives an associative binary operation on
- The following distributivity law holds:

- forms an abelian group with binary operation , inverse map and identity element . This is called the multiplicative group of .