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Mahler measures in a cubic field

Artūras Dubickas (2006)

Czechoslovak Mathematical Journal

We prove that every cyclic cubic extension E of the field of rational numbers contains algebraic numbers which are Mahler measures but not the Mahler measures of algebraic numbers lying in E . This extends the result of Schinzel who proved the same statement for every real quadratic field E . A corresponding conjecture is made for an arbitrary non-totally complex field E and some numerical examples are given. We also show that every natural power of a Mahler measure is a Mahler measure.

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