quaternion algebra

A quaternion algebraPlanetmathPlanetmath over a field K is a central simple algebra over K which is four dimensional as a vector space over K.


  • For any field K, the ring M2×2(K) of 2×2 matrices with entries in K is a quaternion algebra over K. If K is algebraically closedMathworldPlanetmath, then all quaternion algebras over K are isomorphic to M2×2(K).

  • For K=, the well known algebraMathworldPlanetmathPlanetmath of Hamiltonian quaternions is a quaternion algebra over . The two algebras and M2×2() are the only quaternion algebras over , up to isomorphismPlanetmathPlanetmathPlanetmath.

  • When K is a number fieldMathworldPlanetmath, there are infinitely many non–isomorphic quaternion algebras over K. In fact, there is one such quaternion algebra for every even sized finite collection of finite primes or real primes of K. The proof of this deep fact leads to many of the major results of class field theory.

One can show that every quaternion algebra over K other than M2×2(K) is always a division ring.

Title quaternion algebra
Canonical name QuaternionAlgebra
Date of creation 2013-03-22 12:37:54
Last modified on 2013-03-22 12:37:54
Owner djao (24)
Last modified by djao (24)
Numerical id 5
Author djao (24)
Entry type Definition
Classification msc 11R52
Classification msc 16K20