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Quadratic Equations in a Cyclic Number System

Published online by Cambridge University Press:  20 January 2009

R. Wilson
R. Wilson
Affiliation:
University College, Swansea.
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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.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 2 , Issue 3 , January 1931 , pp. 151 - 157
Copyright
Copyright © Edinburgh Mathematical Society 1931

References

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