A field is a set $F$ together with two binary operations addition ($+$) and multiplication ($\cdot$) such that
- Addition and multiplication are both associative;
- Addition and multiplication are both commutative;
- There exist two distinct elements $0\neq 1$ in $F$ such that $0$ is an additive identity and $1$ is a multiplicative identity;
- Every element of $F$ has an additive inverse;
- Every nonzero element of $F$ has a multiplicative inverse;
- Multiplication distributes over addition as follows: $a\cdot (b+c) = (a\cdot b) + (a\cdot c)$ for all $a,b,c\in F$.
Equivalently, a field is a commutative division ring.
Wikidata ID: Q190109