A lower bound for the height in abelian extensions. (English) Zbl 0973.11092

It is shown that if \(\alpha\neq 0\) is an element which is not a root of unity, of an abelian extension of the rationals, then its logarithmic height, \(h(\alpha)\) satisfies \(h(\alpha)\geq(\log 5)/12\). This is used to obtain lower bounds for norms and class-numbers in abelian extensions.


11R06 PV-numbers and generalizations; other special algebraic numbers; Mahler measure
11R18 Cyclotomic extensions
11R20 Other abelian and metabelian extensions
11G50 Heights
11R29 Class numbers, class groups, discriminants
Full Text: DOI


[1] Cassels, J.W.S.; Fröhlich, A., Algebraic number theory, Proceedings of an instructional conference organized by the London mathematical society, (1967), Academic Press London/New York · Zbl 0153.07403
[2] Dobrowolski, E., On a question of Lehmer and the number of irreducible factors of a polynomial, Acta arith., 34, 391-401, (1979) · Zbl 0416.12001
[3] V. Flammang, Mesures de polynômes. Application au diamètre transfini entier, Thèse, Université de Metz.
[4] Laurent, M., Sur la mesure de Mahler de certaines classes d’entiers algébriques, (1980)
[5] Lehmer, D.H., Factorization of certain cyclotomic functions, Ann. of math., 34, 461-479, (1933) · Zbl 0007.19904
[6] Masley, J.M.; Montgomery, H.L., Cyclotomic fields with unique factorization, J. reine angew. math., 286/287, 248-256, (1976) · Zbl 0335.12013
[7] Schinzel, A., On the product of the conjugates outside the unit circle of an algebraic number, Acta arith., 24, 385-399, (1973) · Zbl 0275.12004
[8] Smyth, C.J., On the product of the conjugates outside the unit circle of an algebraic number, Bull. London math. soc., 3, 169-175, (1971) · Zbl 0235.12003
[9] Washington, L.C., Introduction to cyclotomic fields, (1982), Springer-Verlag New York · Zbl 0484.12001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.