Mahler measures generate the largest possible groups (Q1890227)

Let \(\alpha\) be an algebraic number of degree \(d \geq 2\) with minimal polynomial \(P(z) = a_0 z^d + \dots + a_d \in {\mathbb Z}[z]\) over the rationals, and with conjugates \(\alpha_1, \dots, \alpha_d\). The Mahler measure of \(\alpha\) is defined by \(M(\alpha) = | a_0| \prod_{j=1}^d \max(1,| \alpha_j| )\). It is well known that \(\beta = M(\alpha)\) is a positive algebraic integer. Let \({\mathcal M}\) be the set of all Mahler measures of algebraic numbers. The author continues his investigation of \({\mathcal M}\) and related sets by proving that the free multiplicative group generated by \({\mathcal M}\) is the set of all positive algebraic numbers and that the free additive group generated by \({\mathcal M}\) is the set of all real algebraic numbers. More precisely, he shows that every positive algebraic number can be written as the quotient of two elements of \({\mathcal M}\). He also shows that \({\mathcal M}\) is not an additive semigroup. He leaves open the question of whether every real algebraic integer can be expressed as a difference of two elements of \({\mathcal M}\).