Hermite’s Theorem for Function Fields

Hermite’s theorem states that there are only finitely many number fields with bounded discriminant. In this work, we investigate an analog of Hermite’s theorem for function fields: there are only finitely many separable function fields with bounded degree and discriminant. We prove this in the case that the function fields are unramified at ∞. Although Hermite’s theorem for function fields is known through other methods, we used an adaptation of a classical technique from the theory of number fields, namely that of “geometry of numbers”. We expect that the generalization we construct here can, with a few modifications, serve to extend any “geometry of numbers” argument to function fields.