  In this paper we study some fundamental algebraic properties of slice
functions and slice regular functions over an alternative $^*$-algebra $A$ over
$\mathbb{R}$. These recently introduced function theories generalize to higher
dimensions the classical theory of functions of a complex variable. Slice
functions over $A$, which comprise all polynomials over $A$, form an
alternative $^*$-algebra themselves when endowed with appropriate operations.
We presently study this algebraic structure in detail and we confront with
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.
