Field axioms

The field axioms say that a set $F$ with operations $+$ and $\cdot$ satisfies:

1. $F$ is a commutative ring .
2. $F$ satisfies the unital ring axiom with identity $1\neq 0$.
3. Every $a\in F$ with $a\neq 0$ is a unit .

Equivalently, the group of units is $F\setminus\{0\}$. A structure satisfying these axioms is precisely a field .