1. The problem statement, all variables and given/known data Show x^2 + 2 in Z_5[x] is irreducible. This is before the section on the factor theorem (j is a root -> (x-j) is a factor). So I'm not so sure I want to start checking for zero's since "its not available" per se. 2. Relevant equations 3. The attempt at a solution Suppose it was reducible. Then x^2 + 2 = (ax + b) (a^-1x + c). First, I have a feeling that the products are monic, so i dont have to worry about the a, and a^-1. But why would the products would be monic (if it were reducible, even though its not)? I know if a poly can be factored into monics, the poly itself must be monic, (which is clear since the leading coefficient will turn out to be 1 upon collection of like powers.) When I say monic factors, i mean "just" monic factors, with no non-zero coefficients out in front. But I can't say that x^2 + 2 *can be factored into monics* to begin with, since I can't assume this. Please help me with showing why its okay to assume if x^2 + 2 over Z_5 were reducible, it would have monic factors.