## Field Axioms

A field is a setthat is a commutative group with respect to two compatible operations, addition and multiplication, with &quot;compatible&quot; being formalized by distributivity, and the caveat that the additive identity (0) has no multiplicative inverse (one cannot divide by 0).

The most common way to formalize this is by defining a field as a set together with two operations, usually called addition and multiplication, and denoted byandrespectively, such that the following axioms hold;

For all (or more formally,andare binary operations on).
For alland