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Published online by Cambridge University Press: 20 January 2009
Mr D. E. Littlewood has recently discussed the properties of the quadratic equation over the real quaternions and shown that the solutions correspond to the common intersections of four quadrics in four-space. Although complex quaternion solutions may arise, the system of real quaternions to which the coefficients belong is a division algebra. It is of interest, therefore, to discuss the solution of the quadratic when the coefficients are drawn from a system containing divisors of zero.
page 151 note 1 Proc. London Math. Soc., (2), 31 (1930), 40–46.Google Scholar
page 154 note 1 When α is not a divisor of zero and 
, then 
. From (1) this gives x or (ax + 2b) a divisor of zero of the first kind. Four values of u correspond to each case, giving only four solutions other than x adivisor of zero. On the other hand if c1 = c2 = c3, then for 
equations (2) each give 
, whence 
. Similarly 
gives (ax + 2b) a divisor of zero and produces only two values of x.Google Scholar
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