Eugen Netto's Theory of Substitutions
APPLICATIONS TO ALGEBRA.
By DR EUGEN NETTO,
Professor of Mathematics in the University of Giessen.
Revised by the Author and Translated with his Permission
By F N Cole, Ph.D.,
Assistant professor of Mathematics in the University of Michigan.
The presentation of the Theory of Substitutions here given differs in several essential features from that which has heretofore been customary. It will accordingly be proper in this place to state in brief the guiding principles adopted in the present work.
It is unquestionable that the sphere of application of an Algorithm is extended by eliminating from its fundamental principles and its general structure all matters and suppositions not absolutely essential to its nature, and that through the general character of the objects with which it deals, the possibility of its employment in the most varied directions is secured. That the theory of the construction of groups admits of such a treatment is a guarantee for its far-reaching importance and for its future.
If, on the other hand, it is a question of the application of an auxiliary method to a definitely prescribed and limited problem, the elaboration of the method will also have to take into account only this one purpose. The exclusion of all superfluous elements and the increased usefulness of the method is a sufficient compensation for the lacking, but not defective, generality. A greater efficiency is attained in a smaller sphere of action.
The following treatment is calculated solely to introduce in an elementary manner an important auxiliary method for algebraic investigations. By the employment of integral functions from the outset, it is not only possible to give to the Theory of Substitutions, this operating with operations, a concrete and readily comprehended foundation, but also in many cases to simplify the demonstrations, to give the various conceptions which arise a precise form, to define sharply the principal question, and - what does not appear to be least important - to limit the extent of the work.
The two comprehensive treatises on the Theory of Substitutions which have thus far appeared are those of J A Serret and of C Jordan.
The fourth section of the "Algèbre Supérieure" of Serret is devoted to this subject. The radical difference of the methods involved here and there hardly permitted an employment of this highly deserving work for our purposes. Otherwise with the more extensive work of Jordan, the "Traits des substitutions et des équations algébriques." Not only the new fundamental ideas were taken from this book, but it is proper to mention expressly here that many of its proofs and processes of thought also permitted of being satisfactorily employed in the present work in spite of the essential difference of the general treatment. The investigations of Jordan not contained in the "Traité" which have been consulted are cited in the appropriate places.
But while many single particulars are traceable to this "Traité" and to these investigations, nevertheless, the author is indebted to his honoured teacher, L Kronecker, for the ideas which lie at the foundation of his entire work. He has striven to employ to best advantage the benefit which he has derived from the lectures and from the study of the works of this scholarly man, and from the inspiring personal intercourse with him; and he hopes that traces of this influence may appear in many places in his work. One thing he regrets: that the recent important publication of Kronecker, "Grundzüge einer arithmetischen Theorie der algebraischen Grössen," appeared too late for him to derive from it the benefit which he would have wished for himself and his readers.
The plan of the present book is as follows:
In the first part the leading principles of the theory of substitutions are deduced with constant regard to the theory of the integral functions; the analytical treatment retires almost wholly to the background, being employed only at a late stage in reference to the groups of solvable equations.
In the second part, after the establishment of a few fundamental principles, the equations of the second, third and fourth degrees, the Abelian and the Galois equations are discussed as examples. After this follows a chapter devoted to an arithmetical discussion the necessity of which is there explained. Finally the more general, but still elementary questions with regard to solvable equations are examined.
To the preceding I have now to add that the present translation differs from the German edition in many important particulars. Many new investigations have been added. Others, formerly included, which have shown themselves to be of inferior importance, have been omitted. Entire chapters have been rearranged and demonstrations simplified. In short, the whole material which has accumulated in the course of time since the first appearance of the book is now turned to account.
In conclusion the author desires to express his warmest thanks to Mr F N Cole who has disinterestedly assumed the task of translation and performed it with care and skill.
JOC/EFR May 2017
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