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The (arithmetic, or field) norm of a Hurwitz quaternion a + bi + cj + dk, given by a 2 + b 2 + c 2 + d 2, is always an integer. By a theorem of Lagrange every nonnegative integer can be written as a sum of at most four squares. Thus, every nonnegative integer is the norm of some Lipschitz (or Hurwitz) quaternion.
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 ]
Extending the scalars from Q(η) to R (via the standard imbedding), one obtains an isomorphism between the quaternion algebra and the algebra M(2,R) of real 2 by 2 matrices. Choosing a concrete isomorphism allows one to exhibit the (2,3,7) triangle group as a specific Fuchsian group in SL(2, R ) , specifically as a quotient of the modular group .
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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 ]
One chooses a suitable Hurwitz quaternion order in the quaternion algebra, Γ(I) is then the group of norm 1 elements in +. The least absolute value of a trace of a hyperbolic element in Γ( I ) is η 2 + 3 η + 2 {\displaystyle \eta ^{2}+3\eta +2} , corresponding the value 3.936 for the systole of the Klein quartic, one of the highest in this ...
Such functions can be called functions of a quaternion variable just as functions of a real variable or a complex variable are called. As with complex and real analysis , it is possible to study the concepts of analyticity , holomorphy , harmonicity and conformality in the context of quaternions.
In number theory, Hurwitz's theorem, named after Adolf Hurwitz, gives a bound on a Diophantine approximation. The theorem states that for every irrational number ξ there are infinitely many relatively prime integers m , n such that | ξ − m n | < 1 5 n 2 . {\displaystyle \left|\xi -{\frac {m}{n}}\right|<{\frac {1}{{\sqrt {5}}\,n^{2}}}.}