1. Introduction. It seems highly appropriate to us that, in this address, 2 we should present to you results which represent new progress in the structure theory of associative division algebras, a field untouched for nearly thirty years, and which provided the topic of our doctoral dissertation of 1928. As in the paper already published, 8 we shall study the structure of a central division algebra 3), of odd prime degree p over any field ft of characteristic p, which has the property that

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... exists a quadratic extension field $ of ft such that the algebra 25 0 * 2) X $ is cyclic over $. We shall obtain a simplified version of the J A condition that a cyclic algebra £) 0 , of degree p over $, shall possess the factorization property £)o = 3) X $. We shall also derive a new sufficient condition that such a SD shall be cyclic over ft, and shall present a large class of our algebras S)o which satisfy this condition. These results still leave very much open the fundamental question of the existence of noncyclic division algebras of prime degree. However, they do show that we are still far from an end to the consideration of the algebraic aspects of the problem, and are not yet really ready for the computational attack proposed in J A.