In 1947 Skolem proved that the multiplicative group of an algebraic number field K modulo its torsion subgroup is a free abelian group. We outline a proof that this remains true for infinite algebraic extensions of the rationals provided the infinite extension satisfies the Bogomolov property. In contrast to these results, the multiplicative group of all nonzero algebraic numbers modulo its torsion subgroup is known to be a vector space over the rationals, and therefore it is a divisible abelian group.