Remote Access Proceedings of the American Mathematical Society
Green Open Access

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
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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$.


References [Enhancements On Off] (What's this?)

  • [AA+] Mohammed Abouzaid, Jarod Alper, Steve DiMauro, Justin Grosslight, and Derek Smith, Common left- and right-hand divisors of a quaternion integer, J. Pure Appl. Algebra 217 (2013), no. 5, 779-785. MR 3003303, https://doi.org/10.1016/j.jpaa.2012.08.003
  • [CP] Boyd Coan and Cherng-tiao Perng, Factorization of Hurwitz quaternions, Int. Math. Forum 7 (2012), no. 41-44, 2143-2156. MR 2967414
  • [CS] John H. Conway and Derek A. Smith, On quaternions and octonions: their geometry, arithmetic, and symmetry, A K Peters Ltd., Natick, MA, 2003. MR 1957212 (2004a:17002)
  • [H] Adolf Hurwitz, Vorlesungen über die Zahlentheorie der Quaternionen, Verlag von Julius Springer, Berlin, 1919.
  • [KV] Markus Kirschmer and John Voight, Algorithmic enumeration of ideal classes for quaternion orders, SIAM J. Comput. 39 (2010), no. 5, 1714-1747. MR 2592031 (2011c:11171), https://doi.org/10.1137/080734467
  • [P] Cherng-tiao Perng, Factorization of Lipschitz quaternions, preprint, 2011.
  • [S] Jean-Pierre Serre, Local fields, Graduate Texts in Mathematics, vol. 67, Springer-Verlag, New York, 1979. Translated from the French by Marvin Jay Greenberg. MR 554237 (82e:12016)
  • [V] Marie-France Vignéras, Arithmétique des algèbres de quaternions, Lecture Notes in Mathematics, vol. 800, Springer, Berlin, 1980 (French). MR 580949 (82i:12016)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 11R52, 11R27

Retrieve articles in all journals with MSC (2010): 11R52, 11R27


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

American Mathematical Society