Metacommutation of Hurwitz primes

TL;DR

This paper investigates how metacommutation, an operation on Hurwitz primes, induces permutations on primes of a given norm, revealing that the permutation's sign relates to quadratic characters.

Contribution

It characterizes the permutation induced by metacommutation on Hurwitz primes and links its sign to quadratic residues modulo p.

Findings

Permutation sign equals quadratic character of q mod p

Metacommutation induces specific permutations on Hurwitz primes

Provides new insights into algebraic structure of Hurwitz primes

Abstract

Conway and Smith introduced the operation of metacommutation for pairs of primes in the ring of Hurwitz integers in the quaternions. We study the permutation induced on the primes of norm p by a prime of norm q under metacommutation, where p and q are distinct rational primes. In particular, we show that the sign of this permutation is the quadratic character of q modulo p.