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I present this paper for publication to an American journal and in English for the following reason : The theory of linear algebras has been greatly extended through the work of American mathematicians. Of late, German mathematicians have become active in this theory. In particular, they have succeeded in obtaining some apparently remarkable results by using the theory of algebraic numbers, ideals, and abstract algebra, highly developed in Germany in recent decades. These results do not seem to be as well known in America as they should be on account of their importance. This fact is due, perhaps, to the language difference or to the unavailability of the widely scattered sources. This paper develops a new application of the above mentioned theories to the theory of linear algebras. Of particular importance is the fact that purely algebraic results are obtained from deep-lying arithmetical theorems. In the middle part, an account is given of the fundamental algebraic basis for these arithmetical methods. This account is more extended than is necessary for this paper, and should obviate an extended study of several German papers. I am very grateful to Professor F£. T. Engstrom (New Haven) for going through the manuscript and proof with me and anglicising my many literal translations from the German.

transactions of the american mathematical society

Additional note to the author’s: “Theory of cyclic algebras over an algebraic number field” [Trans. Amer. Math. Soc. 34 (1932), no. 1, 171–214; 1501634]