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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.

MSC:

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
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[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
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