Mathematics

Let k be an algebraic number field with degree d over the rational field Q, and discriminant ∆_k. If k has a real embedding then it is easy to prove that k has a generator α such that H(α) ≤ |∆_k|^{1/2d}, where H(α) is the absolute multiplicative Weil height. If k has no real embedding the situation is more complicated. In this case we prove a similar bound on the height of a generator for number fields k such that the Dedekind zeta-function associated to the Galois closure of k/Q satisfies the generalized Riemann hypothesis. This is joint work with Martin Widmer.