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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

   
 

 

Metacommutation of Hurwitz primes


Authors: Henry Cohn and Abhinav Kumar
Journal: Proc. Amer. Math. Soc. 143 (2015), 1459-1469
MSC (2010): Primary 11R52, 11R27
DOI: https://doi.org/10.1090/S0002-9939-2014-12358-6
Published electronically: November 12, 2014
MathSciNet review: 3314061
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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$.


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Additional Information

Henry Cohn
Affiliation: Microsoft Research New England, One Memorial Drive, Cambridge, Massachusetts 02142
Email: cohn@microsoft.com

Abhinav Kumar
Affiliation: Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139
Email: abhinav@math.mit.edu

DOI: https://doi.org/10.1090/S0002-9939-2014-12358-6
Received by editor(s): December 31, 2012
Received by editor(s) in revised form: August 30, 2013
Published electronically: November 12, 2014
Additional Notes: The second author was supported in part by National Science Foundation grants DMS-0757765 and DMS-0952486 and by a grant from the Solomon Buchsbaum Research Fund.
Communicated by: Matthew A. Papanikolas
Article copyright: © Copyright 2014 Henry Cohn and Abhinav Kumar