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

Hurwitz matrix Hurwitz numbers Hurwitz polynomial Hurwitz problem Hurwitz quaternion order Hurwitz quaternion Hurwitz scheme Hurwitz surface Hurwitz zeta...

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...

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

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...

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...

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...

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...

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 {...

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

} 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...

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

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...

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

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

geometry Hermite constant Cubic reciprocity Loewner's torus inequality Hurwitz quaternion Quadratic integer Dixon elliptic functions Both Surányi, László (1997)...

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

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...

Examples Rational number, Real number, Complex number, Quaternions, Octonions Hurwitz quaternion Gaussian integer Theorems and applications Algebraic geometry...

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

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

of twice the dimension.: 45  Hurwitz's theorem (composition algebras) states that the reals, complex numbers, quaternions, and octonions are the only (normed)...

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...

: 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 ∗ (...

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...

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...

of principal congruence subgroups. Here the choices of quaternion algebra and Hurwitz quaternion order are described at the triangle group page. Choosing...

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...

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