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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)].

MSC:

11R27 Units and factorization
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References:

[1] Bernstein, L, (), Lecture Notes in Mathematics
[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; Halter-Koch, F; Stender, H.-J, Unabhängige einheiten für die Körper K = \(Q\)(\(D\^{}\{n\}±dn\)) mit dβdn, (), Abh. math. sem. univ. Hamburg, 50, 162-165, (1980), see also
[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 axn − byn = c und grundeinheiten für einige algebraische zahlkörper von grade n = 3, 4, 6, J. reine angew. math., 290, 24-62, (1977)
[8] Stender, H.-J, Verstümmelte grundeinheiten für biquadratische und bikubische zahlkörper, Math. ann., 232, 55-64, (1978) · Zbl 0372.12009
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