Truncated units. (English) Zbl 0759.11038

Let \(f=X^ n+k_ 1X^{n-1}+\cdots+k_{n-1}X+k_ n\in\mathbb{Z}[X]\) be irreducible, \(f(\omega)=0\), \(D\), \(d\in\mathbb{Z}\), \(D>0\), and \(\varepsilon=1+d^{-1}D\omega-d^{-1}\omega^ 2\). The author provides explicit criteria for \(\varepsilon\) to be an algebraic unit, and he produces tables of examples for \(n=4\) and \(n=6\). Special cases were previously settled by L. Bernstein [Math. Ann. 213, 275-279 (1975; Zbl 0284.12001)].


11R27 Units and factorization
Full Text: DOI


[1] Bernstein, L., (The Jacobi-Perron Algorithm, Its Theory and Application, Vol. 207 (1971), Springer-Verlag: Springer-Verlag New York/Berlin), Lecture Notes in Mathematics · Zbl 0213.05201
[2] Bernstein, L., Truncated units in infinitely many algebraic number fields of degree \(n\) ≥ 4, Math. Ann., 213, 275-279 (1975) · Zbl 0284.12001
[3] Frei, G.; Levesque, C., Independent systems of units in certain algebraic number fields, J. reine angew. Math., 311/312, 116-144 (1979) · Zbl 0409.12006
[4] Halter-Koch, F., Unabhängige Einheitensysteme für eine allgemeine Klasse algebraischer Zahlkörper, Abh. Math. Sem. Univ. Hamburg, 43, 85-91 (1975) · Zbl 0309.12005
[5] Halter-Koch, F.; Stender, H.-J, Unabhängige Einheiten für die Körper \(K = Q(D^n±dn)\) mit \(dβD^n\), Abh. Math. Sem. Univ. Hamburg, 50, 162-165 (1980)
[6] Sprindz̆uk, V. G., “Almost every” algebraic number-field has a large class-number, Acta Arith., 25, 411-413 (1974) · Zbl 0284.12003
[7] Stender, H.-J, Lösbare Gleichungen \(ax^n \) − \( by^n = c\) und Grundeinheiten für einige algebraische Zahlkörper von Grade \(n = 3, 4, 6\), J. reine angew. Math., 290, 24-62 (1977) · Zbl 0499.12004
[8] Stender, H.-J, Verstümmelte Grundeinheiten für biquadratische und bikubische Zahlkörper, Math. Ann., 232, 55-64 (1978) · Zbl 0372.12009
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.