In this paper we give the first construction of a pseudorandom generator, with seed length O(\log n), for \mathrm{CC}_0[p], the class of constant-depth circuits with unbounded fan-in \mathrm{MOD}_p gates, for some prime p. More accurately, the seed length of our generator is O(\log n) for any constant error \epsilon>0. In fact, we obtain our generator by fooling distributions generated by low degree polynomials, over \mathbb{F}_p, when evaluated on the
Boolean cube.This result significantly extends previous constructions that either required a long seed [LVW93] or that could only fool the distribution generated by linear functions over \mathbb{F}_p, when evaluated on the Boolean cube [LRTV09,MZ09].

Enroute of constructing our PRG, we prove two structural results for low degree polynomials over finite fields that can be of
independent interest.

1. Let f be an n-variate degree d polynomial over \mathbb{F}_p.Then, for every \epsilon>0 there exists a subset S \subset [n], whose size depends only on d and \epsilon, such that \sum_{\alpha \in \mathbb{F}_p^n: \alpha \ne 0, \alpha_S=0}|\hat{f}(\alpha)|^2 \leq \epsilon. Namely, there is a constant size subset S such that the total weight of the nonzero Fourier coefficients that do not involve any variable from S is small.

2. Let f be an n-variate degree d polynomial over \mathbb{F}_p.If the distribution of f when applied to uniform zero-one bits is \epsilon-far (in statistical distance) from its distribution when applied to biased bits, then for every \delta>0, f can be approximated, up to error \delta, by a function of a small number (depending only on \epsilon,\delta and d) of lower degree polynomials.