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Every Hurwitz quaternion can be factored as a product of irreducible quaternions. This factorization is not in general unique, even up to units and order, because a positive odd prime p can be written in 24( p +1) ways as a product of two irreducible Hurwitz quaternions of norm p , and for large p these cannot all be equivalent under left and ...
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 ]
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.
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 ]
The (2,3,7) triangle group admits a presentation in terms of the group of quaternions of norm 1 in a suitable order in a quaternion algebra. More specifically, the triangle group is the quotient of the group of quaternions by its center ±1. Let η = 2cos(2π/7). Then from the identity
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 .
perhaps making notes as you read. When you've finished, set aside three hours and write your answers to the questions in Part Three. Whatever your choice, enjoy the journey! THE TURNING POINT The idea started on New Year’s Day in 1980, when my boyfriend (now my husband), Tim, and I woke up in our flat in London. We’d been working
The ring of Hurwitz quaternions, also known as integral quaternions. A quaternion a = a 0 + a 1 i + a 2 j + a 3 k is integral if either all the coefficients a i are integers or all of them are half-integers. All free associative algebras. [1]
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