Journal of Mathematical Sciences

, Volume 224, Issue 6, pp 826–832 | Cite as

The Lengths of the Quaternion and Octonion Algebras


The classical Hurwitz theorem claims that there are exactly four normed algebras with division: the real numbers (ℝ), complex numbers (ℂ), quaternions (ℍ), and octonions (𝕆). The length of ℝ as an algebra over itself is zero; the length of ℂ as an ℝ-algebra equals one. The purpose of the present paper is to prove that the lengths of the ℝ-algebras of quaternions and octonions equal two and three, respectively.


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© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  1. 1.Lomonosov Moscow State University; Moscow Center for Continuous Mathematical EducationMoscowRussia
  2. 2.Lomonosov Moscow State UniversityMoscowRussia

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