
polynomial roots
Two questions that I know the answers to
1. How many polynomials of deg 3 are there in ? The answer is 500.
2. Find the roots of in . Answer= 4,5,1,2
I know for question 2 the 1,2 comes from the fact that its in Z5, but I cant seem to apply that fact to get 1,2 myself. For question 1, what is the methodology of finding 500? I know that each coefficient can only be 04...

For , notice that we only have choices for the coefficient of because it has to be nonzero. We have choices for all the other coefficients, hence we get .
For question , just try all the elements of !