Example

Consider the ring   R = Z/3Z + Z/3Zi,   where i is the square root of -1; so i² = 2. We claim that R is a field.

To see this, suppose that x = a + bi and y = c + di, with a, b, c, d Z/3Z satisfy xy = 0. Multiplying this equation by (a - bi)(c - di), we find

(a² + b²)(c² + d²) = 0.

Both factors are in Z/3Z, which is (a field and hence) a domain. Therefore, at least one of them is zero, say the first (the argument for the second is similar). This means a² = - b², that is, either a = b = 0, or (a/b)² = -1. But there is no element in Z/3Z squaring to -1, so we must have a = b = 0, that is, x = 0.

We conclude that R is a finite domain, whence a field.