A Field

A field (F, +, .) is a set F with operations `+` (or addition) and `.` (or multiplication) satisfying the properties including closure for addition (x + y also in F),
commutativity for addition (x + y = y + x),
associativity for addition (x + (y+z) = (x+y) + z),
existence of additive identity (x + 0 = x),
existence of additive inverse (-x),
closure for multiplication (x.y also in F),
commutativity for multiplication (x.y = y.x),
associativity for multiplication (x.(y.z) = (x.y).z),
existence of multilicative identity (x.1 = x),
existence of multiplicative inverse (1/x), and
distributive law (x.(y+z) = (x.y) + (x.z)).