I've been wondering about doing a cubic analog of the quaternions. The idea is to have two generators Q and R with Q = cbrt(q) and R = cbrt(r), and the non-commutative multiplication rule R*Q = w Q*R (where w = cbrt(1) = (-1 + i sqrt3)/2 = e^(2 pi i/3)). For definiteness, I'm imagining q=2 and r=3. The result seems to be a nine-dimensional space, linear combinations of 1, Q, Q^2, R, R^2, Q R, Q^2 R, Q R^2, Q^2 R^2, with coefficients of the shape a + b w. The type of construction seems to guarantee associativity, needing only to check things like R Q^3 = Q^3 R -- required, since Q^3 is in the ground field which should commute with R, and true, since we pick up three factors of w when we move the R through the Qs while sorting the factors. There are some nice properties like (Q+R)^3 = Q^3 + R^3, which happens because the cross terms like QRR come in three orders and when reordered and collected have coefficient 1+w+w^2 = 0. I haven't proved "no zero divisors", which is an important theorem in the quaternion case. Or worked out the Norm formula, which is needed to compute reciprocals. One proof of the Four Square Theorem (every number is the sum of four squares) uses quternion arithmetic. Perhaps there's a proof of the Nine Cube Theorem lurking somewhere (with q=r=1 ?). Has anyone seen this stuff before? Rich rcs@cs.arizona.edu