I'm trying to prepare for finals, and these questions have me completely stumped. 1. The problem statement, all variables and given/known data 1) For what primes p is x^2 + 1 a factor of x^3 + x^2 + 22x + 15 in F_p[x]? (F_p = finite field with p elements) 2) F a field. Let x^m - 1 have m distinct roots in F, suppose k divides m. Show x^k - 1 has k distinct roots in K. 2. The attempt at a solution 1) Obviously if x^2 + 1 is a factor the other factor must be linear of the form ax + b with coefficients a and b in F_p, a not zero. The only thing I could think of doing is setting up some nasty congruence relations on a and b but they got me nowhere. 2) I don't even know where to start. I don't really see how divisibility plays a role.