A field is a commutative ring $(R,+,\times )$ (henceforth abbreviated simply as $R$, with multiplicative identity $1$ and additive identity $0$) which additionally has the property that every nonzero element has a multiplicative inverse: for every $r\in R$ there is $x\in R$ such that $xr=rx=1$. Conventionally we insist that a field must have more than one element: equivalently, $0\ne 1$.

Examples