Improvement of Nakamula's Upper Bound for the Absolute Discriminant of a Sextic Number Field with Two Real Conjugates

Let K be the compositum of a real quadratic number field K2 and a complex cubic number field K3 . Further, let e be a unit of K which is also a relative unit with respect to K/K2 and K/K3. The absolute discriminant of this non-Galois sextic number field K is estimated from above by a simple, strictly increasing, polynomial function of e . This estimate, which can be used to determine a generator for the cyclic group of relative units, substantially improves a similar bound due to Nakamula. The method employed makes nontrivial use of computer algebra techniques.