Abstract
Let α be an algebraic integer of degreed whosed conjugates are all real. We give a lower bound for the absolute value of conjugates of α in terms ofd and of the number of conjugates outside the interval [−2; 2]. Combining this with a lower bound for Mahler's measure of a polynomial, we obtain a lower bound for the maximal conjugate of a totally real algebraic integer.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
A. Dubickas, On algebraic numbers of small measure,Lith. Math. J.,35, 421–431 (1995).
V. Ennola, Conjugate algebraic integers in an intervalProc. Amer. Math. Soc.,53, 259–261 (1975).
D. H. Lehmer, Factorization of certain cyclotomic functions,Ann. of Math.,34, 461–479 (1933).
R. Louboutin, Sur le mesure de Mahler d'un nombre algébrique,CR Acad. Sci. Paris,296, 707–708 (1983).
R. M. Robinson, Intervals containing infinitely many sets of conjugate algebraic integers, in:Studies in Mathematical Analysis and Related Topics, Stanford (1962), pp. 305–315.
A. Schinzel and H. Zassenhaus, A refinement of two theorems of Kronecker,Michigan Math. J.,12, 81–85 (1965).
I. Schur, Über die Verteilung der Wurzeln bei gewissen algebraischen Gleichungen mit ganzzahligen Koeffizienten,Math. Zeitsch.,1, 377–402 (1918).
T. Zaimi, Minoration du diamètre d'un entier algébrique totalement réel,CR Acad. Sci. Paris,319, 417–419 (1994).
Additional information
Vilnius University, Naugarduko 24, 2006 Vilnius, Lithuania. Published in Lietuvos Matematikos Rinkinys, Vol. 37, No. 1, pp. 18–25, January–March, 1997.
Rights and permissions
About this article
Cite this article
Dubickas, A. On the maximal conjugate of a totally real algebraic integer. Lith Math J 37, 13–19 (1997). https://doi.org/10.1007/BF02465435
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF02465435