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Keywords:
Mahler measure; Pisot numbers; cubic extension
Summary:
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.
References:
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