In this paper we study some fundamental algebraic properties of slice functions and slice regular functions over an alternative ∗ ^* -algebra A A over R \mathbb {R} . These recently introduced function theories generalize to higher dimensions the classical theory of functions of a complex variable. Slice functions over A A , which comprise all polynomials over A A , form an alternative ∗ ^* -algebra themselves when endowed with appropriate operations. We presently study this algebraic structure in detail and we confront questions about the existence of multiplicative inverses. This study leads us to a detailed investigation of the zero sets of slice functions and of slice regular functions, which are of course of independent interest.