Fix a finite field, over which our objects are defined. We will start by discussing a classical way to compute the number of degree d monic square-free polynomials, using the zeta function of the affine line. Then we will discuss how to compute the same number, using the machinery of etale cohomology, treating many steps as a blackbox.

Both methods "generalize" to compute the number of degree d monic square-free polynomials with non-vanishing conditions, but one of the answers is "better" than the other for computational purposes.

This is a graduate version of the talk I gave for Math Club on September 22 with the same title. Speaker(s): GilYoung Cheong (UM)