  From a unifying lemma concerning fusion rings, we prove a collection of
number-theoretic results about fusion, braided, and modular tensor categories.
First, we prove that every fusion ring has a dimensional grading by an
elementary abelian 2-group. As a result, we bound the order of the
multiplicative central charge of arbitrary modular tensor categories. We also
introduce Galois-invariant subgroups of the Witt group of nondegenerately
braided fusion categories corresponding to algebraic number fields generated by
Frobenius-Perron dimensions. Lastly, we provide a complete description of the
fields generated by the Frobenius-Perron dimensions of simple objects in
$\mathcal{C}(\mathfrak{g},k)$, the modular tensor categories arising from the
representation theory of quantum groups at roots of unity, as well as the
fields generated by their Verlinde eigenvalues.
