Some explicit upper bounds on the class number and regulator of a cubic field with negative discriminant (Q1105641)

Let \(K\) be a cubic number field with regulator \(R\) and discriminant \(\Delta <0\); then \(\Delta = df^2\), where \(d\) is the discriminant of a quadratic field. Let \(h\) be the number of ideal classes of the ring of integers of \(K\). The authors obtain explicit bounds on the product \(hR\), in terms of \(d\) and \(f\). For example, if \(K\) is pure cubic then, \(hR<(2f\cdot \log (f)+f\cdot \log (3))/6\), and if, in addition, \(\Delta <-6912\), then \(hR>0.44(| \Delta | /27)^{1/4}/\log (| \Delta | /27)\).