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.

DOI : 10.5802/jtnb.529
Toufik Zaimi 1

1 King Saud University Dept. of Mathematics P. O. Box 2455 Riyadh 11451, Saudi Arabia
@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},
     mrnumber = {2212134},
     zbl = {05016596},
     language = {en},
     url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.529/}
}
TY  - JOUR
AU  - Toufik Zaimi
TI  - The cubics which are differences of two conjugates of an algebraic integer
JO  - Journal de théorie des nombres de Bordeaux
PY  - 2005
SP  - 949
EP  - 953
VL  - 17
IS  - 3
PB  - Université Bordeaux 1
UR  - https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.529/
DO  - 10.5802/jtnb.529
LA  - en
ID  - JTNB_2005__17_3_949_0
ER  - 
%0 Journal Article
%A Toufik Zaimi
%T The cubics which are differences of two conjugates of an algebraic integer
%J Journal de théorie des nombres de Bordeaux
%D 2005
%P 949-953
%V 17
%N 3
%I Université Bordeaux 1
%U https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.529/
%R 10.5802/jtnb.529
%G en
%F JTNB_2005__17_3_949_0
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 | Zbl

[2] A. Dubickas, C. J. Smyth, Variations on the theme of Hilbert’s Theorem 90. Glasg. Math. J. 44 (2002), 435–441. | MR | Zbl

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

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

[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 | Zbl

Cité par Sources :