Characterizations of quadratic, cubic, and quartic residue matrices

We construct a collection of matrices defined by quadratic residue symbols, termed “quadratic residue matrices”, associated to the splitting behavior of prime ideals in a composite of quadratic extensions of $\mathbb{Q}$, and prove a simple criterion characterizing such matrices. We also study the analogous classes of matrices constructed from the cubic and quartic residue symbols for a set of prime ideals of $\mathbb{Q}(\sqrt{-3})$ and $\mathbb{Q}(i)$, respectively.