Certain quartic fields with small regulators
The author gives explicit lower bounds of the regulators \(R_K\) of quartic fields \(K\) in terms of the discriminant of \(K\) and his results cover quartic fields of unit ranks 1, 2 and 3. Using the fields \(\mathbb{Q}(\omega)\), where \(\omega\) is a root of \(x^4- sx^3+ (t+ 2u) x^2- usx+ 1\), the author shows that his estimates are in a sense best possible by exhibiting for certain values of \(s\), \(t\) and \(u\) fundamental systems of units. The above polynomial looks like a generalization of the polynomial \(x^4- sx^3+ rx^2- usx+ 1\), with \(r= ((s^2- mz^2)/4)+ zu\), considered by \textit{G. Gras} [Publ. Math. Fac. Sci. Besançon, Theor. Nombres, Exp. 5 (1979; Zbl 0472.12009)]. This paper will certainly prove interesting to number theorists involved with units (and class numbers).
lower bounds of the regulators of quartic fields
scientific article; zbMATH DE number 874227