The cubics which are differences of two conjugates of an algebraic integer
Toufik Zaimi1
1King Saud University Dept. of Mathematics P. O. Box 2455 Riyadh 11451, Saudi Arabia
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 
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
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