The following exercise is from Golan's book on linear algebra.
Problem: Consider the algebra of polynomials over $GF(7)$, the field with 7 elements.
a) Find a nonzero polynomial such that the corresponding polynomial function is identically equal to zero.
b) Is the polynomial $6x^4+3x^3+6x^2+2x+5$ irreducible?
Work so far: The first part is easy. The polynomial $x^7-x$ works by Fermat's little theorem. The second part is trickier. If the polynomial is reducible, it facts into the product of a linear term and something else, or it factors as two quadratics. The first case is easy to exclude; simply plug all seven elements of $\mathbb{Z}_7$ into the polynomial and confirm none of them are a root. The second is harder. Of course, one could just set up the systems of equations resulting from
$(ax^2+bx+c)(dx^2+ex+f)$
and go through all the possible values of $a,c,d,f$ and see if the resulting values of $b$ and $e$ are permissible, and while I know that would eventually give me the answer, I have no desire to do all of those computations. Is there a slicker way?