Algebras and their arithmetics

257 Pages·1938·13.27 MB·English

by  Leonard Eugene Dickson

Algebras and their arithmetics, by Leonard Eugene Dickson ...
Dickson, Leonard E. (Leonard Eugene), 1874-1954.
New York, G. E. Stechert & co., 1938.
http://hdl.handle.net/2027/mdp.39015042078389
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ALGEBRAS AND THEIR
ARITHMETICS
By
Leonard Eugene Dickson
Professor of Mathematics, University of Chicago
REPRINT OF THE I923 EDITION
NEW YORK
G. E. STECHERT & CO.
1938
Copyright By
I923
The University of Chicago
Att Rights Reserved
Pubtished Juty I923
^5' i 7
<f
Y.fc/^, PREFACE
a.-:
The chief purpose of this book is the development for
the first time of a general theory of the arithmetics of
algebras, which furnishes a direct generalization of the
classic theory of algebraic numbers. The book should
appeal not merely to those interested in either algebra
or the theory of numbers, but also to those interested in
the foundations of mathematics. Just as the final
stage in the evolution of number was reached with
the introduction of hypercomplex numbers (which make
up a linear algebra), so also in arithmetic, which began
with integers and was greatly enriched by the introduc
tion of integral algebraic numbers, the final stage of its
development is reached in the present new theory of
arithmetics of linear algebras.
Since the book has interest for wide classes of readers,
no effort has been spared in making the presentation
clear and strictly elementary, requiring on the part of
the reader merely an acquaintance with the simpler
parts of a first course in the theory of equations. Each
definition is illustrated by a simple example. Each
chapter has an appropriate introduction and summary.
The author's earlier brief book, Linear Algebras
(Cambridge University Press, 1914), restricted attention
to complex algebras. But the new theory of arithmetics
of algebras is based on the theory of algebras over a
general field. The latter theory was first presented by
Wedderburn in his memoir in the Proceedings of the
London Mathematical Society for 1907. The proofs of
viii PREFACE
some of his leading theorems were exceedingly com
plicated and obscured by the identification of algebras
having the same units but with co-ordinates in different
fields. Scorza in his book, Corpi Numerici e Algebre
(Messina [1021], ix+462 pp.), gave a simpler proof of
the theorem on the structure of simple algebras, but
omitted the most important results on division algebras
as well as the principal theorem on linear algebras.
An outline of a new simpler proof of that theorem was
placed at the disposal of the author by Wedderburn,
with whom the author has been in constant correspond
ence while writing this book, and who made numerous
valuable suggestions after reading the part of the manu
script which deals with the algebraic theory. However,
many of the proofs due essentially to Wedderburn have
been recast materially. Known theorems on the rank
equations of complex algebras have been extended by
the author to algebras over any field. The division
algebras discovered by him in 1906 are treated more
simply than heretofore.
Scorza's book has been of material assistance to the
author although the present exposition of the algebraic
part differs in many important respects from that by
Scorza and from that in the author's earlier book.
But the chief obl1gations of the author are due to
Wedderburn, both for his invention of the general theory
of algebras and for his cordial co-operation in the present
attempt to perfect and simplify that theory and to
render it readily accessible to general readers.
The theory of arithmetics of algebras has been sur
prisingly slow in its evolution. Quite naturally the
arithmetic of quaternions received attention first;
PREFACE ix
the initial theory presented by Lipschitz in his book of
1886 was extremely complicated, while a successful
theory was first obtained by Hurwitz in his memoir
of 1896 (and book of 1919). Du Pasquier, a pupil of
Hurwitz, has proposed in numerous memoirs a definition
of integral elements of any rational algebra which is
either vacuous or leads to insurmountable difficulties
discussed in this book. Adopting a new definition, the
author develops at length a far-reaching general theory
whose richness and simplicity mark it as the proper
generalization of the theory of algebraic numbers to the
arithmetic of any rational algebra.
Acknowledgments are due to Professor Moore, the
chairman of the Editorial Committee of the University
of Chicago Science Series, for valuable suggestions
both on the manuscript and on the proofsheets of the
chapter on arithmetics.
L. E. D1ckson
Univers1ty op Chicago
June,
1923