Metacommutation of Hurwitz primes
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- by Henry Cohn and Abhinav Kumar
- Proc. Amer. Math. Soc. 143 (2015), 1459-1469
- DOI: https://doi.org/10.1090/S0002-9939-2014-12358-6
- Published electronically: November 12, 2014
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
- 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, DOI 10.1016/j.jpaa.2012.08.003
- Boyd Coan and Cherng-tiao Perng, Factorization of Hurwitz quaternions, Int. Math. Forum 7 (2012), no. 41-44, 2143–2156. MR 2967414
- 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
- Adolf Hurwitz, Vorlesungen über die Zahlentheorie der Quaternionen, Verlag von Julius Springer, Berlin, 1919.
- Markus Kirschmer and John Voight, Algorithmic enumeration of ideal classes for quaternion orders, SIAM J. Comput. 39 (2010), no. 5, 1714–1747. MR 2592031, DOI 10.1137/080734467
- Cherng-tiao Perng, Factorization of Lipschitz quaternions, preprint, 2011.
- Jean-Pierre Serre, Local fields, Graduate Texts in Mathematics, vol. 67, Springer-Verlag, New York-Berlin, 1979. Translated from the French by Marvin Jay Greenberg. MR 554237
- Marie-France Vignéras, Arithmétique des algèbres de quaternions, Lecture Notes in Mathematics, vol. 800, Springer, Berlin, 1980 (French). MR 580949
Bibliographic Information
- Henry Cohn
- Affiliation: Microsoft Research New England, One Memorial Drive, Cambridge, Massachusetts 02142
- MR Author ID: 606578
- ORCID: 0000-0001-9261-4656
- Email: cohn@microsoft.com
- Abhinav Kumar
- Affiliation: Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139
- MR Author ID: 694441
- Email: abhinav@math.mit.edu
- 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
- © Copyright 2014 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
- MathSciNet review: 3314061