On some cubic or quartic algebraic units
Let \(\varepsilon\) be an algebraic unit that is not a complex root of unity. Assume that the rank of the unit group of the order \(\mathbb Z[\varepsilon]\) is equal to one. Then \(\mathbb Q(\varepsilon)\) is either (i) a real quadratic number field, (ii) a not totally real cubic number field, or (iii) a totally imaginary quartic number field. The subject of the paper is to answer the question: is \(\varepsilon\) a fundamental unit of \(\mathbb Z[\varepsilon]\)? The answer for the case (ii) is in [\textit{T. Nagell}, Acta Math. 55, 33--65 (1930; JFM 56.0168.04)]; a new proof of it is given in [\textit{S. Louboutin}, J. Number Theory 121, No. 1, 30--39 (2006; Zbl 1171.11057)]. In [\textit{J. Ramanujan}, J. Ramanujan Math. Soc. 23, No. 2, 191--210 (2008; Zbl 1165.11076)] the present author gave the answer for the case (i). In [\textit{S. M. Park} and \textit{G. N. Lee}, J. Number Theory 129, No. 6, 1338--1349 (2009; Zbl 1167.11040)] the answer for (iii) is given. In the paper under review, there is no new result, but the author gives a simpler proof of the key result for the cases (ii) and (iii).
scientific article; zbMATH DE number 5690731