PREFACE.
It has been my endeavor in this book to lead by easy stages a
reader, entirely unacquainted with the subject, to an appreciation
of some of the fundamental conceptions in the general theory of
algebraic numbers. With this object in view, I have treated the
theory of rational integers more in the manner of the general
theory than is usual, and have emphasized those properties of
these integers which find their analogues in the general theory.
The same may be said of the general quadratic realm, which has
been treated rather as an example of the general realm of the
nth degree than simply as of the second degree, as little use as
was possible, without too great sacrifice of simplicity, being made
of the special properties of the quadratic realm in the proofs.
The theorems and their proofs have therefore been so formulated
as to be readily extendable, in most cases, to the general realm
of the nth degree, and it is hoped that a student, who wishes to
continue the study of the subject, will find the reading of works
on the general theory, such as Hilbert's Bericht iiber die Theorie
der Algebraischen Zahlkorper, rendered easier thereby. The
realm ^(V — i) has been discussed at some length with two
objects in view; first, to show how exactly the theorems relating
to rational integers can be carried over to the integers of a higher
realm when once the unique factorization theorem has been
established ; and second, to illustrate, by a brief account of Gauss' work
in biquadratic residues, the advantage gained by widening our field
of operation. The proofs of the theorems relating to biquadratic
residues have necessarily been omitted but the examples given will
make the reader acquainted with their content. The realms
fe(V — 3) and k{y/2) have been briefly discussed in order to
introduce the reader to modifications which must be made in our
conceptions of integers and units. In k(^/ — 5), the failure of
the unique factorization law is shown and its restoration in terms
of ideal factors is foreshadowed.