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 QQ, 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 Q(−3) and Q(i)Q(i), respectively.