The theory of numbers, by Robert D. Carmichael ...

CHAPTER VI

OTHER TOPICS
~ 40. INTRODUCTION
THE theory of numbers is a vast discipline and no single
volume can adequately treat of it in all of its phases. A short
book can serve only as an introduction; but where the field
is so vast such an introduction is much needed. That is the
end which the present volume is intended to serve; and it
will best accomplish this end if, in addition to the detailed theory
already developed, some account is given of the various directions in which the matter might be carried further.
To do even this properly it is necessary to limit the number
of subjects considered. Consequently we shall at once lay
aside many topics of interest which would find a place in an
exhaustive treatise. We shall say nothing, for instance, about
the vast domain of algebraic numbers, even though this is one
of the most fascinating subjects in the whole field of mathematics. Consequently, we shall not refer to any of the extensive theory connected with the division of the circle into equal
parts. Again, we shall leave unmentioned many topics connected with the theory of positive integers; such, for instance,
is the frequency of prime numbers in the ordered system of
integers-a subject which contains in itself an extensive and
elegant theory.
In ~~ 41-44 we shall speak briefly of each of the following
topics: theory of quadratic residues, Galois imaginaries, arithmetic forms, analytical theory of numbers. Each of these alone
would require a considerable volume for its proper development.
All that we can do is to indicate the nature of the problem in
each case and in some cases to give a few of the fundamental
results.
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