  Modular categories are important algebraic structures in a variety of
subjects in mathematics and physics. We provide an explicit, motivated and
elementary definition of a modular category over a field of characteristic 0 as
an equivalence class of solutions to a set of polynomial equations. We conclude
that within each class of solutions, there is one which consists entirely of
algebraic numbers. These algebraic solutions make it possible to discuss
defining algebraic number fields of modular categories and their Galois twists.
One motivation for such a definition is an arithmetic theory of modular
categories which plays an important role in their classification. Another is to
facilitate implementation of computer-based tools to resolve computational and
classification problems intractible by other means. We observe some basic
properties of Galois twists of modular categories and make conjectures about
their relation to the the intrinsic data of modular categories.
