On the non-Archimedean metric Mahler measure

Recently, Dubickas and Smyth constructed and examined the metric Mahler measure and the metric na\"ive height on the multiplicative group of algebraic numbers. We give a non-Archimedean version of the metric Mahler measure, denoted $M_\infty$, and prove that $M_\infty(\alpha) = 1$ if and only if $\alpha$ is a root of unity. We further show that $M_\infty$ defines a projective height on $\bar{\mathbb Q}^\times/ \bar{\mathbb Q}^\times_\mathrm{tors}$ as a vector space over $\mathbb Q$. Finally, we demonstrate how to compute $M_\infty(\alpha)$ when $\alpha$ is a surd.