Metacommutation of Hurwitz primes

Abstract

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.

Metacommutation as a Group Action on the Projective Line Over 𝔽_{𝕡}

“…The following proposition establishes this correspondence. (The proof we give follows the argument in [2]. )…”

“…While the characterization of the metacommutation mapping as a group action on C p , as [2] gives, is simple, H is still not as intuitive as we want. So in the spirit of simplifying our study of metacommutation, we recall the following proposition.…”

“…As in [2], we say this factorization of h is modelled on the factorization of N (h) = p 1 p 2 · · · p n . When N (h) is square-free, we call such a factorization unique up to unit migration, because if the p i are distinct, then every factorization of h modelled on…”

“…We call such a factorization unique up to recombination. As in [2], we say a left ideal HP for prime P ∈ H lies over a rational prime p if N (P ) = p.…”

“…The metacommutation map on primes of norm p sends every P to P ′ . For each Q, this map induces a permutation of the Hurwitz primes of norm p. In [2], Cohn and Kumar prove that the sign of this permutation is q p , where · p is the Legendre symbol, and that this permutation has 1 + tr(Q) 2 − 4q p fixed points. Although the current literature gives several properties of the permutation the metacommutation map induces, it does not directly describe this permutation.…”