Norms extremal with respect to the Mahler measure and a generalization of Dirichlet's unit theorem

In this thesis, we introduce and study several norms constructed to satisfy an extremal property with respect to the Mahler measure. These norms naturally generalize the metric Mahler measure introduced by Dubickas and Smyth. We show that bounding these norms on a certain subspace implies Lehmer's conjecture and in at least one case that the converse is true as well. We evaluate these norms on a class of algebraic numbers that include Pisot and Salem numbers, and for surds. We prove that the infimum in the construction is achieved in a certain finite dimensional space for all algebraic numbers in one case, and for surds in general, a finiteness result analogous to that of Samuels and Jankauskas for the t-metric Mahler measures. Next, we generalize Dirichlet's S-unit theorem from the usual group of S-units of a number field K to the infinite rank group of all algebraic numbers having nontrivial valuations only on places lying over S. Specifically, we demonstrate that the group of algebraic S-units modulo torsion is a Q-vector space which, when normed by the Weil height, spans a hyperplane determined by the product formula, and that the elements of this vector space which are linearly independent over Q retain their linear independence over R.