Characterizations of quadratic, cubic, and quartic residue matrices

TL;DR

This paper introduces and characterizes matrices derived from quadratic, cubic, and quartic residue symbols, linking their properties to the splitting behavior of primes in specific number field extensions.

Contribution

It constructs residue matrices from quadratic, cubic, and quartic symbols and provides a criterion to characterize these matrices based on prime splitting behavior.

Findings

Characterization criterion for quadratic residue matrices

Extension of residue matrix concepts to cubic and quartic cases

Insights into prime splitting in quadratic, cubic, and quartic extensions

Abstract

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.