If F is a finite field, F[x] contains infinitely many primes

Let F be a finite field. Prove that F[x] contains infinitely many prime polynomials.

Suppose to the contrary that F[x] contains only finitely many primes, and label these p_1,p_2,\ldots,p_n. Let q = \prod p_i, and consider q+1. Because F[x] is a Euclidean domain, it is a unique factorization domain. Note that each p_i has degree at least 1, since the constant polynomials in F[x] are units. Thus q+1 has degree at least 1, and in particular is not a unit or zero. Thus q+1 can be written as a product of irreducibles in F[x], which (again because F[x] is a UFD) are precisely the p_i. Suppose q+1 = p_kd. Then 1 = p_k(d - q^\prime), and in particular p_k is a unit- a contradiction.

Thus F[x] contains infinitely many primes.

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