Title: Counting number fields by their smallest defining polynomial
Abstract: When do two irreducible polynomials with integer coefficients define the same number field? Improving on work of Bhargava, Shankar, and Wang, we show that in a certain statistical sense, this usually only happens if the polynomials lie in the same orbit of a particular action of GL_2 x GL_1. We use this to count number fields whose smallest defining polynomial has coefficients of size at most X.