I will discuss some recent joint work with T. Tao on the distribution of degree d polynomials f(X1, .., XN ) with coefficients in some fixed finite field k (such as F2 or F3). If the values taken by such a polynomial are not close to being equidistributed then it turns out that f must be expressible in terms of lower degree polynomials, if the characteristic of k is large enough. When the characteristic becomes very small (and in particular if k = F2) these issues become a lot less clear and remain unsolved in some cases. I will also discuss a counterexample to the so called ”Inverse Gowers Conjecture” in finite characteristic. I’ll also describe what can be salvaged from the conjecture, and explain why this counterexample does not leave us unduly worried about our programme to count solutions to linear equations in primes.