Improvement of Nakamula’s upper bound for the absolute discriminant of a sextic number field with two real conjugates

Abstract:

Let K be the compositum of a real quadratic number field ${{\mathbf {K}}_2}$ and a complex cubic number field ${{\mathbf {K}}_3}$. Further, let $\varepsilon$ be a unit of K which is also a relative unit with respect to ${\mathbf {K}}/{{\mathbf {K}}_2}$ and ${\mathbf {K}}/{{\mathbf {K}}_3}$. The absolute discriminant of this non-Galois sextic number field K is estimated from above by a simple, strictly increasing, polynomial function of $\varepsilon$. 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.