The maximal modulus of an algebraic integer

Author:
David W. Boyd

Journal:
Math. Comp. **45** (1985), 243-249, S17

MSC:
Primary 11R06

DOI:
https://doi.org/10.1090/S0025-5718-1985-0790657-8

MathSciNet review:
790657

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Abstract: The maximal modulus of an algebraic integer is the absolute value of its largest conjugate. We compute the minimum of the maximal modulus of all algebraic integers of degree *d* which are not roots of unity, for *d* at most 12. The computations suggest that the minimum is never attained for a reciprocal algebraic integer. The truth of this conjecture would show that the conjecture of Schinzel and Zassenhaus follows from a theorem of Smyth. We further test our conjecture by computing the minimum of the maximal modulus of all reciprocal algebraic integers of degree *d* which are not roots of unity, for *d* at most 16. Our computations strongly suggest that the best constant in the conjecture of Schinzel and Zassenhaus is 1.5 , where is the smallest P.V. number. They also shed some light on a recent conjecture of Lind concerning the Perron numbers.

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Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1985-0790657-8

Keywords:
Algebraic integer,
maximal modulus,
Schinzel-Zassenhaus conjecture,
Perron numbers,
Smyth's theorem,
Newton's formulas

Article copyright:
© Copyright 1985
American Mathematical Society