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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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