## Field Axioms

A field is a set that 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 by and respectively, such that the following axioms hold;

Closure of under addition and multiplication
For all  (or more formally, and are binary operations on ).
For all  and For all  There exists an element of called the additive identity element and denoted by 0, such that for all Likewise, there is an element, called the multiplicative identity element and denoted by 1, such that for all The additive identity and the multiplicative identity may not be the same.
For every there exists such that Similarly, for any other than 0, there exists an element such that (The elements and are denoted and respectively.) In other words subtraction and division operations exist.
For all  Examples: are all fields as is for n prime. The tables below are for a finite field with four elements. 0 1 A B 0 1 A B 0 0 0 0 0 0 0 1 A B 1 0 1 A B 1 1 0 B A A 0 A B 1 A A B 0 1 B 0 B A 1 B B A 1 0 