In mathematics, a Hurwitz quaternion (or Hurwitz integer) is a quaternion whose components are either all integers or all half-integers (halves of odd integers; a mixture of integers and half-integers is excluded). The set of all Hurwitz quaternions is
That is, either a, b, c, d are all integers, or they are all half-integers.
H is closed under quaternion multiplication and addition, which makes it a subring of the ring of all quaternions H. Hurwitz quaternions were introduced by Adolf Hurwitz (1919).
A Lipschitz quaternion (or Lipschitz integer) is a quaternion whose components are all integers. The set of all Lipschitz quaternions
forms a subring of the Hurwitz quaternions H. Hurwitz integers have the advantage over Lipschitz integers that it is possible to perform Euclidean division on them, obtaining a small remainder.
Both the Hurwitz and Lipschitz quaternions are examples of noncommutative domains which are not division rings.
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In mathematics, a Hurwitzquaternion (or Hurwitz integer) is a quaternion whose components are either all integers or all half-integers (halves of odd...
The Hurwitzquaternion order is a specific order in a quaternion algebra over a suitable number field. The order is of particular importance in Riemann...
Euclidean Hurwitz algebras, of which the quaternions are the largest associative algebra (and hence the largest ring). Further extending the quaternions yields...
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...
include the twenty-four Hurwitzquaternions that have the norm 1 and form vertices of a 24-cell polychoron. Hamilton defined a quaternion as the quotient of...
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 {...
E(α) of conjugation α(⋅)α by the Hurwitzquaternion α = m + ni + pj + qk restricted to the subspace of quaternions spanned by i, j, k, which is given...
of principal congruence subgroups. Here the choices of quaternion algebra and Hurwitzquaternion order are described at the triangle group page. Choosing...
Algebraic integer Cyclotomic field Eisenstein integer Eisenstein prime Hurwitzquaternion Proofs of Fermat's theorem on sums of two squares Proofs of quadratic...
} One chooses a suitable Hurwitzquaternion order Q H u r {\displaystyle {\mathcal {Q}}_{\mathrm {Hur} }} in the quaternion algebra, Γ(I) is then the...
In mathematics, quaternions are a non-commutative number system that extends the complex numbers. Quaternions and their applications to rotations were...
by taking the unit groups of orders in quaternion algebras over number fields (for example the Hurwitzquaternion order). Similar constructions can be performed...
mathematical objects, such as polynomials, quadratic integers and Hurwitzquaternions. In the latter cases, the Euclidean algorithm is used to demonstrate...
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...
: 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 ∗ (...
{\displaystyle D} (see Hurwitzquaternion order), described explicitly by Noam Elkies [1]. In order to construct the first Hurwitz triplet, consider the...
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