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Reciprocal Algebraic Integers Whose Mahler Measures are Non-Reciprocal

Published online by Cambridge University Press:  20 November 2018

David W. Boyd
David W. Boyd*
Affiliation:
Department of Mathematics University of British Columbia Vancouver, B.C., CanadaV6T 1Y4
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Abstract

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The Mahler measure M (α) of an algebraic integer α is the product of the moduli of the conjugates of α which lie outside the unit circle. A number α is reciprocal if α- 1 is a conjugate of α. We give two constructions of reciprocal a for which M (α) is non-reciprocal producing examples of any degree n of the form 2h with h odd and h ≥ 3, or else of the form with s ≥ 2. We give explicit examples of degrees 10, 14 and 20.

Keywords

Primary 12A10Secondary 12A55
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 30 , Issue 1 , 01 March 1987 , pp. 3 - 8
Copyright
Copyright © Canadian Mathematical Society 01

References

1. Beaumont, R.A and Peterson, R.P, Set-transitive permutation groups, Canad. Jour. Math. 7 (1955), pp. 35–42.Google Scholar
2. Boyd, D.W, Inverse problems for Mahler's measure, in Diophantine Analysis, ed. Loxton, J.H and der Poorten, A.J. van, Camb. Univ. Press, 1986.Google Scholar
3. Boyd, D.W, Perron units which are not Mahler measures, Ergodic Th. and Dyn. Syst., to appear.Google Scholar
4. Boyle, M., Pisot, Salem and Perron numbers in ergodic theory and topological dynamics, xeroxed notes, November 1982.Google Scholar
5. Mignotte, M., Sur les conjugués des nombres de Pisot, C.R. Acad. Sci. Paris 298(1984), p. 21.Google Scholar
6. Smyth, C.J, On the product of the conjugates outside the unit circle of an algebraic integer, Bull. Lond. Math. Soc. 3 (1971), pp. 169–175.Google Scholar
7. Soicher, L. and McKay, J., Computing Galois groups over the rationals, Jour. Number Theory 20 (1985), pp. 273–281.Google Scholar
8. Watson, G.N, Singular moduli (4), Acta Arith. 1 (1935), pp. 284–323.Google Scholar
9. Weber, H., Lehrbuch der Algebra, Bd II, Braunschweig 1908, reprinted by Chelsea, N.Y., 1961.Google Scholar
10. Yamamoto, Y., On unramified Galois extensions of quadratic number fields, Osaka J. Math. I (1970), pp. 57–76.Google Scholar