The existence of small defining polynomials for number fields

Abstract

We prove that every number field of degree $n$ and absolute discriminant $D$ has a defining integral polynomial of degree $n$ whose coefficients are all $O_n(\sqrt D)$ in size, thus resolving a 1998 conjecture of Ruppert. Our proof compares ideal lattices with their trace duals, via which we handle all number fields with the exception of CM fields whose maximal totally real subfields have small discriminant. We then treat the remaining fields by averaging completed Epstein zeta functions over all possible ideal classes and archimedean scalings, for which we make critical use of Hecke's formula and the subconvexity theorem of Michel and~Venkatesh. Our proof does not use the Quasi-Riemann Hypothesis.