Algebraic field

by Patrick Stevens Jun 13 2016 updated Jun 14 2016

A field is a structure with addition, multiplication and division.

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