Foundations of the Theory of Algebraic Invariants - ISBN:0000000000

Foundations of the Theory of Algebraic Invariants

Date: 21 April 2011, 03:29

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The theory of algebraic invariants has found insufficient attention in
Russian mathematical literature. The book by Alekseev [I I]*, written
in 1899, is largely out of date, while individual chapters in certain text-
books on algebra (Sushkevich, Bocher, etc.) written later, give only
the beginnings of the theory. The present book is to fill this gap. Its
essential special feature is wide utilization of classical methods as well
as of the basic concepts and notation of tensor algebra; this makes it
possible to present all problems at once in as general a form as possible.
In addition, the Author believes that only by this means can one succeed
in bringing full clarity to the problem of Aronhold's symbolic (?18).
Throughout I have attempted not to leave out of sight the close link
which exists between invariant theory and geometry; the extensive
geometric introduction of Chapter 1 also serves this purpose. Chapter
11 which is concerned with the foundations of tensor algebra also bears
an introductory character. The general propositions of the theory of
algebraic invariants are given in Chapters III and IV, the most important
particular results in Chapters V-VII; among these Chapter V is more
classical in spirit.
At the end of each section, I have given exercises of which there are
altogether approximately 500. They serve a double purpose in that some
explain the preceding work, others consider problems treated insuffi-
ciently in the main text. At times, a set of problems will present some
substantial section of theory, for example, the exercises following Ё 3,
10-14, 25 contain the classification of binary forms of fourth order in
the real domain. Certain problems relate to the subsequent text; in that
case they have been provided with asterisks.
In collecting the exercises, I have made use of the references [3, 11,
13, 14, 16, 19] as well as of H. Beck's ,Koordinaten Geometric". How-
ever, a substantial number of the problems have been published here
for the first time.
In conclusion, I wish to express my gratitude to Ia. S. Dubnov, who
has studied the manuscript of Chapter I and made a number of valuable
comments, and to M. G. Freidinii for his painstaking editorial work.
G. B. Gurevich.

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