Zaïmi, Toufik The cubics which are differences of two conjugates of an algebraic integer. (English) Zbl 1153.11336 J. Théor. Nombres Bordx. 17, No. 3, 949-953 (2005). Summary: We show that a cubic algebraic integer over a number field \(K\), with zero trace is a difference of two conjugates over \(K\) of an algebraic integer. We also prove that if \(N\) is a normal cubic extension of the field of rational numbers, then every integer of \(N\) with zero trace is a difference of two conjugates of an integer of \(N\) if and only if the 3-adic valuation of the discriminant of \(N\) is not 4. Cited in 1 Document MSC: 11R04 Algebraic numbers; rings of algebraic integers 11R16 Cubic and quartic extensions Keywords:cubic algebraic integers; normal cubic extension of the field of rational numbers PDF BibTeX XML Cite \textit{T. Zaïmi}, J. Théor. Nombres Bordx. 17, No. 3, 949--953 (2005; Zbl 1153.11336) Full Text: DOI Numdam Numdam EuDML OpenURL References: [1] A. Dubickas, On numbers which are differences of two conjugates of an algebraic integer. Bull. Austral. Math. Soc. 65 (2002), 439-447. · Zbl 1028.11065 [2] A. Dubickas, C. J. Smyth, Variations on the theme of Hilbert’s Theorem 90. Glasg. Math. J. 44 (2002), 435-441. · Zbl 1112.11308 [3] S. Lang, Algebra. Addison-Wesley Publishing, Reading Mass. 1965. · Zbl 0193.34701 [4] A. Schinzel, Selected Topics on polynomials. University of Michigan, Ann Arbor, 1982. · Zbl 0487.12002 [5] T. Zaimi, On numbers which are differences of two conjugates over \(\mathbb{Q}\)of an algebraic integer. Bull. Austral. Math. Soc. 68 (2003), 233-242. · Zbl 1043.11073 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.