Number Theory Seminar: Guillermo Mantilla-Soler (UBC)

Given a nonzero integer d and a positive integer n we know, by Hermite's Theorem, that there exist only finitely many degree n number fields of discriminant d. It is thus natural to ask whether there are refinements of the discriminant which completely determine the isomorphism class of a number field. In this talk we will consider the integral trace form as such refinement. By using one of Bhargava's composition of cubes, we show that the integral trace form is a complete invariant for cubic fields with positive fundamental discriminant. If time allows we will discuss some further results for higher degree number fields.