A field is any set \(F\) obeying the following field axioms:
- Addition is a binary operation
- Multiplication is a binary operation
- Addition is a commutative operation
- Multiplication is a commutative
- Addition is associative
- Multiplication is associative
- There is an additive identity in \(F\)
- There is a multiplicative identity in \(F\)
- There exists additive inverses in \(F\)
- There exits multiplicative inverses in \(F\)
- Multiplication distributes over addition
Some examples include the set of rational numbers \(\mathbb{Q}\), the set of integers \(\mathbb{Z}\) and the set of two elements \(\{0,1\}\) where addition and multiplication are taken modulo 2.