Algebraic number fields generated by Frobenius-Perron dimensions in fusion rings

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.