Let be a finite field. Prove that contains infinitely many prime polynomials.
Suppose to the contrary that contains only finitely many primes, and label these . Let , and consider . Because is a Euclidean domain, it is a unique factorization domain. Note that each has degree at least 1, since the constant polynomials in are units. Thus has degree at least 1, and in particular is not a unit or zero. Thus can be written as a product of irreducibles in , which (again because is a UFD) are precisely the . Suppose . Then , and in particular is a unit- a contradiction.
Thus contains infinitely many primes.