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The Hurwitz quaternions form an order (in the sense of ring theory) in the division ring of quaternions with rational components. It is in fact a maximal order ; this accounts for its importance. The Lipschitz quaternions, which are the more obvious candidate for the idea of an integral quaternion , also form an order.
As well as being an elegant theorem in its own right, Lagrange's four square theorem has useful applications in areas of mathematics outside number theory, such as combinatorial design theory. The quaternion-based proof uses Hurwitz quaternions, a subring of the ring of all quaternions for which there is an analog of the Euclidean algorithm.
The Hurwitz quaternion order is a specific order in a quaternion algebra over a suitable number field. The order is of particular importance in Riemann surface theory, in connection with surfaces with maximal symmetry , namely the Hurwitz surfaces . [ 1 ]
Adolf Hurwitz (German: [ˈaːdɔlf ˈhʊʁvɪts]; 26 March 1859 – 18 November 1919) was a German mathematician who worked on algebra, analysis, geometry and number theory. Early life [ edit ]
Thus, the quaternion components ,,, are either all integers or all half-integers, depending on whether is even or odd, respectively. The set of Hurwitz quaternions forms a ring; that is to say, the sum or product of any two Hurwitz quaternions is likewise a Hurwitz quaternion.
Hurwitz and Frobenius proved theorems that put limits on hypercomplexity: Hurwitz's theorem says finite-dimensional real composition algebras are the reals , the complexes , the quaternions , and the octonions , and the Frobenius theorem says the only real associative division algebras are , , and .