In mathematics, a Hurwitz quaternion (or Hurwitz integer) is a quaternion whose components are either all integers or all half-integers (halves of odd...
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Hurwitz matrix Hurwitz numbers Hurwitz polynomial Hurwitz problem Hurwitz quaternion order Hurwitz quaternion Hurwitz scheme Hurwitz surface Hurwitz zeta...
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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. The (arithmetic...
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Euclidean Hurwitz algebras, of which the quaternions are the largest associative algebra (and hence the largest ring). Further extending the quaternions yields...
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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...
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In mathematics, a quaternion algebra over a field F is a central simple algebra A over F that has dimension 4 over F. Every quaternion algebra becomes a...
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Versor (redirect from Unit quaternion)
include the twenty-four Hurwitz quaternions that have the norm 1 and form vertices of a 24-cell polychoron. Hamilton defined a quaternion as the quotient of...
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(2,3,7) triangle group (section Quaternion algebra)
ij = −ji. One chooses a suitable Hurwitz quaternion order Q H u r {\displaystyle {\mathcal {Q}}_{\mathrm {Hur} }} in the quaternion algebra. Here the order Q...
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In group theory, the quaternion group Q8 (sometimes just denoted by Q) is a non-abelian group of order eight, isomorphic to the eight-element subset {...
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E(α) of conjugation α(⋅)α by the Hurwitz quaternion α = m + ni + pj + qk restricted to the subspace of quaternions spanned by i, j, k, which is given...
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For example, the Hurwitz quaternions form a maximal order in the quaternions with rational co-ordinates; they are not the quaternions with integer coordinates...
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complex numbers, the quaternions, or the octonions, and that there are no other possibilities. Such algebras, sometimes called Hurwitz algebras, are examples...
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Klein quartic (section Quaternion algebra construction)
} One chooses a suitable Hurwitz quaternion order Q H u r {\displaystyle {\mathcal {Q}}_{\mathrm {Hur} }} in the quaternion algebra, Γ(I) is then the...
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1009, 1025, 1459, 1537, 2091, ... (sequence A179982 in the OEIS) Hurwitz quaternion order Elkies, N.: Shimura curve computations. Algorithmic number theory...
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center of the other). They form a ring called the Hurwitz quaternion ring. The 24 Hurwitz quaternions of norm 1 form the vertices of a 24-cell centered...
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Algebraic integer Cyclotomic field Eisenstein integer Eisenstein prime Hurwitz quaternion Proofs of Fermat's theorem on sums of two squares Proofs of quadratic...
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geometry Hermite constant Cubic reciprocity Loewner's torus inequality Hurwitz quaternion Quadratic integer Dixon elliptic functions Both Surányi, László (1997)...
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In mathematics, quaternions are a non-commutative number system that extends the complex numbers. Quaternions and their applications to rotations were...
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which lie at distance 1 from the origin. Viewed as quaternions, these are the unit Hurwitz quaternions. The 24-cell has unit radius and unit edge length...
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Examples Rational number, Real number, Complex number, Quaternions, Octonions Hurwitz quaternion Gaussian integer Theorems and applications Algebraic geometry...
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by taking the unit groups of orders in quaternion algebras over number fields (for example the Hurwitz quaternion order). Similar constructions can be performed...
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mathematical objects, such as polynomials, quadratic integers and Hurwitz quaternions. In the latter cases, the Euclidean algorithm is used to demonstrate...
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Cayley–Dickson construction (category Historical treatment of quaternions)
of twice the dimension.: 45 Hurwitz's theorem (composition algebras) states that the reals, complex numbers, quaternions, and octonions are the only (normed)...
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Composition algebra (redirect from Para-Hurwitz algebra)
advanced the study of the Hurwitz problem with a survey of efforts to that date, and by exhibiting the method of doubling the quaternions to obtain Cayley numbers...
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noncommutative subring of the quaternions, hence a noncommutative domain. Similarly, the set of all Hurwitz quaternions, that is, quaternions of the form a + b i...
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: 484 In dimensions 4 and 8 these are para-quaternion and para-octonion algebras.: 40, 41 A para-Hurwitz algebra satisfies: 48 ( x ∗ y ) ∗ x = x ∗ (...
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of principal congruence subgroups. Here the choices of quaternion algebra and Hurwitz quaternion order are described at the triangle group page. Choosing...
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structure F 4 {\displaystyle \mathrm {F_{4}} } forms the ring of Hurwitz quaternions that is represented by the vertices of the 24-cell — with this regular...
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None of the three is a complex algebra. Hurwitz quaternions form a non-commutative *-ring with the quaternion conjugation. The matrix algebra of n × n...
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{\displaystyle D} (see Hurwitz quaternion order), described explicitly by Noam Elkies [1]. In order to construct the first Hurwitz triplet, consider the...
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