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, $~$0 \not = 1$~$.