The cubics which are differences of two conjugates of an algebraic integer
Journal de Théorie des Nombres de Bordeaux, Volume 17 (2005) no. 3, pp. 949-953.

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.

On montre qu’un entier algébrique cubique sur un corps de nombres K, de trace nulle est la différence de deux conjugués sur K d’un entier algébrique. On prouve aussi que si N est une extension cubique normale du corps des rationnels, alors tout entier de N de trace zéro est la différence de deux conjugués d’un entier de N si et seulement si la valuation 3-adique du discriminant de N est différente de 4.

Published online:
DOI: 10.5802/jtnb.529
Toufik Zaimi 1

1 King Saud University Dept. of Mathematics P. O. Box 2455 Riyadh 11451, Saudi Arabia
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Toufik Zaimi. The cubics which are differences of two conjugates of an algebraic integer. Journal de Théorie des Nombres de Bordeaux, Volume 17 (2005) no. 3, pp. 949-953. doi : 10.5802/jtnb.529. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.529/

[1] A. Dubickas, On numbers which are differences of two conjugates of an algebraic integer. Bull. Austral. Math. Soc. 65 (2002), 439–447. | MR: 1910496 | 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. | MR: 1956551 | Zbl: 01890767

[3] S. Lang, Algebra. Addison-Wesley Publishing, Reading Mass. 1965. | MR: 197234 | Zbl: 0193.34701

[4] A. Schinzel, Selected Topics on polynomials. University of Michigan, Ann Arbor, 1982. | MR: 649775 | Zbl: 0487.12002

[5] T. Zaimi, On numbers which are differences of two conjugates over of an algebraic integer. Bull. Austral. Math. Soc. 68 (2003), 233–242. | MR: 2016300 | Zbl: 1043.11073

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