In mathematics, a Hurwitz quaternion (or Hurwitz integer) is a quaternion whose components are either all integers or all half-integers (halves of odd...

Euclidean Hurwitz algebras, of which the quaternions are the largest associative algebra (and hence the largest ring). Further extending the quaternions yields...

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...

determinant Hurwitz-stable matrix Routh–Hurwitz matrix Hurwitz numbers Hurwitz polynomial Hurwitz problem Hurwitz quaternion order Hurwitz quaternion Hurwitz scheme...

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...

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...

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 {...

1009, 1025, 1459, 1537, 2091, ... (sequence A179982 in the OEIS) Hurwitz quaternion order Elkies, N.: Shimura curve computations. Algorithmic number theory...

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...

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...

In mathematics, quaternions are a non-commutative number system that extends the complex numbers. Quaternions and their applications to rotations were...

} 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...

mathematical objects, such as polynomials, quadratic integers and Hurwitz quaternions. In the latter cases, the Euclidean algorithm is used to demonstrate...

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...

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...

For example, the Hurwitz quaternions form a maximal order in the quaternions with rational co-ordinates; they are not the quaternions with integer coordinates...

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...

complex numbers, the quaternions, or the octonions, and that there are no other possibilities. Such algebras, sometimes called Hurwitz algebras, are examples...

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...

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...

algebra Pre-Lie algebra Poisson algebra Process algebra Quadratic algebra Quaternion algebra Rees algebra Relation algebra Relational algebra Rota–Baxter algebra...

unique factorization property. The ring of Hurwitz quaternions, also known as integral quaternions. A quaternion a = a0 + a1i + a2j + a3k is integral if...

group of units of the Hurwitz quaternions, which has order 24, contains a normal subgroup of order 8 isomorphic to the quaternion group, and is the binary...

Algebraic integer Cyclotomic field Eisenstein integer Eisenstein prime Hurwitz quaternion Proofs of Fermat's theorem on sums of two squares Proofs of quadratic...

by taking the unit groups of orders in quaternion algebras over number fields (for example the Hurwitz quaternion order). Similar constructions can be performed...

of the following: R (the real numbers) C (the complex numbers) H (the quaternions) These algebras have real dimension 1, 2, and 4, respectively. Of these...

smallest abstract groups that are not any symmetry group in 3D, are the quaternion group (of order 8), Z3 × Z3 (of order 9), the dicyclic group Dic3 (of...

half-integers is not allowed). This lattice is isomorphic to the lattice of Hurwitz quaternions. The root system G2 has 12 roots, which form the vertices of a hexagram...

{\displaystyle D} (see Hurwitz quaternion order), described explicitly by Noam Elkies [1]. In order to construct the first Hurwitz triplet, consider the...

geometry Hermite constant Cubic reciprocity Loewner's torus inequality Hurwitz quaternion Quadratic integer Dixon elliptic functions Equianharmonic Both Surányi...