Give an example of integral domain having infinite number of elements, yet of finite characteristic?

Question

Taken from Herstein, Ring Theory, Problem 7, Page 130

Answer 1

A standard example is the ring of all polynomials of 1 variable over a field of finite characteristic - let F_{q} = GF(q) is a Galois Field with a finite characteristic q /q can be only a degree of a prime number/. The ring

F_{q}[x] = GF(q)[x] of all (infinitely many) polynomials with coefficients, taken from GF(q) is the required example (it is well-known that it is an integral domain):

Char(GF(q)[x]) = Char(GF(q)) = q

For instance, the smallest possible field GF(2) = Z_{2} = {0, 1} leads to

GF(2)[x] with characteristic 2: if we take, say

x² + x + 1, then 2 * (x² + x + 1) = (1+1)x² + (1+1)x + (1+1) = 0,

since over GF(2) we have 1+1 = 0.

Similarly the field of the rational fractions (field of ratios) of GF(q)[x] is an example of an infinite FIELD with a finite characteristic q.