  This work studies slice functions over finite-dimensional division algebras.
Their zero sets are studied in detail along with their multiplicative inverses,
for which some unexpected phenomena are discovered. The results are applied to
prove some useful properties of the subclass of slice regular functions,
previously known only over quaternions. Firstly, they are applied to derive
from the maximum modulus principle a version of the minimum modulus principle,
which is in turn applied to prove the open mapping theorem. Secondly, they are
applied to prove, in the context of the classification of singularities, the
counterpart of the Casorati-Weierstrass theorem.
