On commutative linear algebras in which division is always uniquely possible

1. We consider commutative linear algebras in 2n units, with coordinates in a general field F, such that n of the units define a sub-algebra forming a field F( J). The elements of the algebra may be exhibited compactly in the form A + BJ, where A and B range over F(J). As multiplication is not associative in general, A and B do not play the role of coordinates, so that the algebra is not binary in the usual significance of the term.t Nevertheless, by the use of this binary niotation, we may exhibit in a very luminous forml the multiplicatioln-tables of certain algebras in four and six unlits, given in an earlier paper. i Proof of the existence of the algebras and of the uniqtueness of division niow presents no difficulty. The form of the corresponding algebra in 2n units becomes obvious. After thus perfecting and extending known results, we attack the problem of the deternmination of all algebras with the prescribed properties. An extensive new class of algebras is obtained.