Prime factorization of integral Cayley octaves. (English) Zbl 0830.17017

As observed by Coxeter, there is a maximal order of integral octaves (the even unimodular lattice of dimension 8) admitting also some division with remainder. Unlike the associative case, however, it seems to have been unknown how to use this property for computing common divisors. But in this paper, the author develops such an algorithm and, as a consequence, he obtains unique prime factorization for integral octaves. This result allows him to give an algebraic proof of Jacobi’s formula for sums of eight squares.


17D05 Alternative rings
11R52 Quaternion and other division algebras: arithmetic, zeta functions
11E25 Sums of squares and representations by other particular quadratic forms
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