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The Lengths of the Quaternion and Octonion Algebras

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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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Correspondence to A. E. Guterman.

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Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 453, 2016, pp. 22–32.

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Guterman, A.E., Kudryavtsev, D.K. The Lengths of the Quaternion and Octonion Algebras. J Math Sci 224, 826–832 (2017).

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