A field is a special set.

## constituents

- distinct elements of at least \(0\) and \(1\)
- operations of addition and multiplication

## requirements

- closed
- commutativity
- associativity
- identities (both additive and multiplicative)
- inverses (both additive and multiplicative)
- distribution

Therefore, \(\mathbb{R}\) is a field, and so is \(\mathbb{C}\) (which we proved in properties of complex arithmetic).

## additional information

Main difference between group: there is *one* operation is group, a field has two operations.