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

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.

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.

@article{JTNB_2005__17_3_949_0,
     author = {Toufik Zaimi},
     title = {The cubics which are differences of two conjugates of an algebraic integer},
     journal = {Journal de Th\'eorie des Nombres de Bordeaux},
     pages = {949--953},
     publisher = {Universit\'e Bordeaux 1},
     volume = {17},
     number = {3},
     year = {2005},
     doi = {10.5802/jtnb.529},
     zbl = {05016596},
     mrnumber = {2212134},
     language = {en},
     url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.529/}
}
Toufik Zaimi. The cubics which are differences of two conjugates of an algebraic integer. Journal de Théorie des Nombres de Bordeaux, Tome 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