We give, for a fixed prime number p, necessary conditions for the existence of products of p+1 distinct monic irreducible polynomials in one variable over the finite field F_p, each raised to an arbitrary positive power, that are even perfect polynomials; i.e., they have at least one root in F_p, and they are equal to the sum of all their monic divisors.As a consequence, we prove that the set of such polynomials is finite, and if q = (p−1)/2 is also prime, so that q is a Sophie Germain prime and p is a safe prime, then it is empty. This is the first known finiteness result for perfect polynomials. We might consider it as an analogue of Dickson’s result that proves the finiteness of the set of odd perfect numbers with a fixed number of distinct prime divisors, each raised to an arbitrary positive power.