The construction of all division algebras is the outstanding problem in the theory of linear algebras. In the history of this problem, the procedure has been to examine first the necessary and sufficient conditions that the constructed algebra be associative; and second, the conditions that it be a division algebra. Since L. E. Dickson's announcement in 1905 of his discovery of a system of non-commutative division algebras, he has published three papers examining associativity conditions for such algebras. In the first of these papers, algebras associated with cyclic equations were discussed; in the second, general results were obtained for the abelian non-cyclic case, and for that type of non-abelian case where the Galois group of the basic equation was a solvable group. This study was carried forward by Williamson, in a paper in which detailed associativity conditions were established for the general case of a two generator group, and for a significant special case of the three generator group. In a third paper in 1930, Dickson gave a simplification of his previous method by which he reached specific results for the general case of the two and three generator problems. The second line of investigation, the study of the conditions under which a given associative algebra is a division algebra, has been advanced by Dickson, Wedderburn, and Albert. In addition to these two aspects of the problem, there is a third phase which is concerned with normal division algebras. It is known that any division algebra can be normalized by a suitable extension of the field, so that its order is the square of an integer, and a third line of study has established that all division algebras of orders four, nine, or sixteen are among those for which associativity conditions have been found.
The present paper continues the investigation of associativity conditions, by considering the algebras related to an equation whose Galois group has four independent generators. Using Dickson's most recent simplification of his previous method, we set up necessary and sufficient conditions that these algebras be associative. Since the method is inductive, all the results which Dickson obtained for the two or three generator cases will be presupposed. The notation of the present paper is the same as in Dickson's except that v, y, z, w, ε, δ of his paper are replaced by z1, w1, z2, w2, δ1, δ2 respectively. Numbered theorems and formulas in square brackets refer to theorems and formulas in his paper.