On Hypercomplex Numbers

The reader is referred to the following papers on this subject:-Dickson 2, 3. * I.e., isomorphism with regard both to addition and multiplication.G 2 * An algebra is said to have a modulus e, if e is an element such that ex = x = xe for every element x of A. * Epsteen (4), p. 444.t Scheffera (13).* In other words, if e u e 3 , ..., e a ia a basis of A, e y x, e^v, ..., e a x are necessarily independent if Ax = A. t Since, if A" = B + C, then B + C-= A°-> = A' => B + C, and therefore G -C 3 (mod £ ) .* Cartan (1), p. 57.* It is here assumed that rational elements which are independent in F are also independent in F. * This gives a proof of a theorem by Allan to the effect that the order of a primitiTe algebra is of the form btfi.

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