## 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
For all
There exists an element ofcalled the additive identity element and denoted by 0, such that for allLikewise, there is an element, called the multiplicative identity element and denoted by 1, such that for allThe additive identity and the multiplicative identity may not be the same.
For everythere existssuch thatSimilarly, for any other than 0, there exists an elementsuch that(The elements andare denotedandrespectively.) In other words subtraction and division operations exist.