We consider the space of all degree-$n$ integer polynomials, ordered by the sizes of their coefficients, where $n\geq 3$ is odd. I will first describe joint work with Bhargava and Wang in which we compute the density of such polynomials having squarefree discriminant (in particular, proving that such a density exists!) Next, I will explain joint work with Ho and Varma, in which we prove (using the previous result) that for every odd integer n>1, there exist infinitely many degree-n S_n fields with odd class number.