Sebastian Monnet

S4-quartics with prescribed norms

Let K be a number field with \mathbb{Q}-basis \{e_1, ..., e_n\}, and let \alpha be a rational number. It is natural to ask whether the “norm equation” N_{K/\mathbb{Q}}(x_1e_1 + ... + x_ne_n) = \alpha has rational solutions. Since the answer depends only on K, we may ask how often this norm equation has rational solutions as we vary K. The case of abelian number fields was solved by Frei-Loughran-Newton, and in this talk we present one of the simplest non-abelian cases: S_4-quartics.