Two Recent Books on Algebras

theory, the relation of hypercomplex numbers to quadric surfaces, abelian integrals, etc. Moreover, as will be seen below, the book would have gained both in power and charm if the second part had been compressed. Part I, on the general theory of number fields (130 pages) is divided into three chapters. Of these, the first gives the definition of a number field via four postulates, the rudiments of the theory and the basic differences between finite and infinite fields. In this chapter he also introduces the notation for ^-tuples and gives the elements of the theory of matrices and determinants. The second chapter is the longest of the three and is devoted to polynomials. After the usual definitions for polynomials in several variables, he gives the orthodox work on divisibility, rational fractions, the zeros of a polynomial and the solution of a system of linear equations together with Leibniz's formula for a general field (modular or otherwise), Galois' generalization of Fermat's theorem and Euclid's algorithm for the highest common factor. In the final chapter, he gives the fundamental properties of Galois fields. The reader is soon struck by the fact that the author seems to want every definition and theorem expressed with a meticulous care that is rare. All through the text he makes precise statements similar to this : "Polynomials in the same field and in the same indeterminates that differ at most by null monomials are regarded as not distinct". At first, such precise care will please the cautious mathematician; but, after page upon page of such detailed precision, the reader begins to weary. In the case of some theorems, this results in his spreading over a page or more a proof that could be given in a few lines with complete rigor and greater simplicity. In fact, one is inclined to wonder if this style would not tend to dull the interest of the young student for whom the text is written. Part II, on the general theory of linear algebras (273 pages), is divided into six chapters, of which the first gives in forty-seven pages not only the orthodox elementary work with matrices, but also much interesting material not usually found in a text-book (such as Kowalewski's Determinantentheorie) of which the most important are Cayley's theorem that every square matrix satisfies its characteristic equation, Frobenius' beautiful theorem that the equation of lowest degree satisfied by a square matrix is a factor of the characteristic determinant obtained in a certain definite manner and Hadamard's well known result for the maximum value of a determinant. The remaining chapters of this part are devoted to the general theory of algebras and look much like a revision of the first part of the author's memoir Le algèbre di ordine qualunque e le matrici di Miemann which appeared in the EENDICONTI DI PALEBMO for 1921. In many sections, the point of view and the method are those of Wedderburn's beautiful 1924.] TWO BOOKS ON ALGEBRAS 269 memoir in the PROCEEDINGS OF THE LONDON SOCIETY for 1908, though Professor Scorza has polished a number of the proofs—some to a slight degree and some to a greater degree, notably the proof of the theorem on the structure of simple algebras. Although Chapter 2 is, with one exception, the longest in the book, it contains scarcely more than a long array of definitions and minor results in the general theory of linear algebras. To one already familiar with the essentials of the theory, the significance of this chapter will, of course, be clear; but the reviewer wonders if, when these multitudinous minor results are presented at the beginning of the theory, in a terminology unnecessarily complicated, they may not confuse or discourage some students. Chapters 3 and 4 are brief and discuss properties of an algebra connected with its invariant subalgebras and idempotent elements, respectively. Of these, the most important is the theorem that any algebra is the sum of its maximum nilpotent subalgebra and a semi-simple algebra (page 255). Chapter 5 is also comparatively brief and is concerned with the coordinates of an algebra, characteristic determinants, etc. The last chapter is the longest, occupying nearly one fifth of the entire book. In this he proves the set of theorems on the structure of the different types of algebras: simple algebras, matric algebras, division algebras; but ignores the very interesting and important work on division algebras by Dickson and Wedderburn referred to above in connection with Dickson's book. Although the subject-matter of these five chapters is essentially the same as that of the first half of Dickson's book, it would be scarcely possible to imagine two mathematical books more different in their treatment. This difference is partly due to the unnecessary detail in Scorza's presentation, and partly due to the unfortunate fact that Scorza uses the older terminology which has been discarded. Also, it sometimes seems to the reader that he delights in multiplying the number of technical terms. For example, since his algebra eccezionale of an algebra A is simply the maximum nilpotent invariant subalgebra of A, why not call it such ? The latter term is not only more descriptive, but is also the usual one. At the end he has placed a bibliography of fifty-three books and memoirs on number fields, matrices and linear algebras. Although this is surely far better than none, it is not all that might be desired, since there are several errors, both of omission and commission. For example, why does it omit Steinitz, Ernst, Algebraïsche Theorie der Körper, JOURNAL FÜR MATHEMATIK, vol. 137 (1909), pp. 167-309 ; Dickson, L. E., Linear associative algebras and abelian equations, TRANSACTIONS OF THIS SOCIETY, vol. 15 (1914), pp. 31-46; Wedderburn, J. H. M., A type of primitive algebra, ibid., pp, 162-166? 270 OLIVE c. HAZLETT [May-June, The last two articles (together with a brief but suggestive paper by Wedderburn in the TRANSACTIONS for 1921 which, presumably, appeared too late to be included in the bibliography) are, without doubt, the most interesting and the only important articles on the general theory of division algebras, and contain all that has been discovered about this difficult and extremely fascinating branch of linear algebras since Wedderburn's memoir in the PROCEEDINGS OF THE LONDON SOCIETY mentioned elsewhere in this review. One notes with dismay that there is no index. This lack is an inconvenience in any book of this size and is only slightly ameliorated by putting in bold-face type the caption of every section and of every definition. In fact, one might almost say that the presence or lack of an index is a characteristic invariant which distinguishes AngloAmerican texts from Continental ones.* But altogether, the book contains a great deal of information not previously available outside of technical periodicals ; and Professor Scorza is to be congratulated on the courage with which he attempted and the care with which he finished the task that he had set himself. As one reads, one can not but feel that the task has been to him a pleasant one and that when it was completed, he left it with a caress.