Multiplication on Rn: 1. Division algebra and vector product

Datta, Basudeb (1998) Multiplication on Rn: 1. Division algebra and vector product Resonance - Journal of Science Education, 3 (1). pp. 8-14. ISSN 0971-8044

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Official URL: http://www.ias.ac.in/resonance/Jan1998/pdf/Jan1998...

Related URL: http://dx.doi.org/10.1007/BF02838637

Abstract

In 1843 Hamilton discovered the quaternions and in the same year Graves found an algebra with 8 basis elements. So, mathematicians knew the division algebra structures (over R) on R, R2, R4 and R8 in the first half of the 19th century itself. It took more than 100 years to prove that these are all the division algebras over R. This follows from a theorem of Adams (which we will discuss in Part 2). In this part we also discuss 'vector products' on R3 and R7 and Hurwitz's theorem on 'sums of squares formulae'.

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