The cubics which are differences of two conjugates of an algebraic integer

Toufik Zaimi[1]

  • [1] King Saud University Dept. of Mathematics P. O. Box 2455 Riyadh 11451, Saudi Arabia

Journal de Théorie des Nombres de Bordeaux (2005)

  • Volume: 17, Issue: 3, page 949-953
  • ISSN: 1246-7405

Abstract

top
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 .

How to cite

top

Zaimi, Toufik. "The cubics which are differences of two conjugates of an algebraic integer." Journal de Théorie des Nombres de Bordeaux 17.3 (2005): 949-953. <http://eudml.org/doc/249429>.

@article{Zaimi2005,
abstract = {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 $4.$},
affiliation = {King Saud University Dept. of Mathematics P. O. Box 2455 Riyadh 11451, Saudi Arabia},
author = {Zaimi, Toufik},
journal = {Journal de Théorie des Nombres de Bordeaux},
keywords = {cubic algebraic integers; normal cubic extension of the field of rational numbers},
language = {eng},
number = {3},
pages = {949-953},
publisher = {Université Bordeaux 1},
title = {The cubics which are differences of two conjugates of an algebraic integer},
url = {http://eudml.org/doc/249429},
volume = {17},
year = {2005},
}

TY - JOUR
AU - Zaimi, Toufik
TI - The cubics which are differences of two conjugates of an algebraic integer
JO - Journal de Théorie des Nombres de Bordeaux
PY - 2005
PB - Université Bordeaux 1
VL - 17
IS - 3
SP - 949
EP - 953
AB - 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 $4.$
LA - eng
KW - cubic algebraic integers; normal cubic extension of the field of rational numbers
UR - http://eudml.org/doc/249429
ER -

References

top
  1. A. Dubickas, On numbers which are differences of two conjugates of an algebraic integer. Bull. Austral. Math. Soc. 65 (2002), 439–447. Zbl1028.11065MR1910496
  2. A. Dubickas, C. J. Smyth, Variations on the theme of Hilbert’s Theorem 90. Glasg. Math. J. 44 (2002), 435–441. Zbl1112.11308MR1956551
  3. S. Lang, Algebra. Addison-Wesley Publishing, Reading Mass. 1965. Zbl0193.34701MR197234
  4. A. Schinzel, Selected Topics on polynomials. University of Michigan, Ann Arbor, 1982. Zbl0487.12002MR649775
  5. T. Zaimi, On numbers which are differences of two conjugates over ℚ of an algebraic integer. Bull. Austral. Math. Soc. 68 (2003), 233–242. Zbl1043.11073MR2016300

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.