SECTIONS. 10-17 18-24 V THE ALGEBRAIC EQUATION By G. A. MILLER I. GENERAL INTRODUCTION 1. Aim of the monograph. The present monograph aims to give a sketch of some of the most fundamental processes in which the algebraic equation occupies a central position, and thus to fix the attention more completely on the underlying thoughts and the historical setting than would be feasible in a short treatise on the theory of equations. The monograph is intended to supplement such treatises rather than to replace them. By means of the historic setting of many elementary facts it is hoped that parts of it may be useful also to those who have only such a knowledge of the equation as would naturally result from an elementary course in algebra. 2. How it should be read. The reader is advised not to insist on understanding every statement before proceeding to the next. To some readers such concepts as domain of rationality, substitution group, and p-valued rational function may be new, and our short account of them may not appear entirely satisfactory. A slight knowledge of such dominating concepts and of their applications is, however, much better than total ignorance, and if the present monograph leads to an intelligent search for knowledge along these important lines its perusal will not have been in vain. 3. Mathematics presupposed. To avoid prolixity it has seemed desirable to presuppose, in a few places, an elementary knowledge of determinants as well as a knowledge of the first 211 derivative of a function of a single variable. As it seemed undesirable to presuppose an elementary knowledge of the Galois theory of equations some fundamental processes could not be sketched with the completeness that would be desirable. It is hoped, however, that the viewpoint which has been adopted will tend to prepare the way for this general theory; this is especially true of the methods used to solve the cubic and the biquadratic equations. While the common road to a knowledge of the equation leads through numerous problems, it is sometimes desirable to take a broad survey of the historic setting and of the underlying principles, and thus to gather new inspiration and a deeper insight. It is hoped that the present monograph may aid in taking such surveys. = 4. Type of questions studied. Equations of the form x" = 1 play an important rôle in the general theory of equations. Since the fundamental properties of these equations are treated in Monograph No. VII, secs. 28, 29, they are not given in the present monograph. As the roots of the equation_x"= ±a may be obtained by multiplying those of x"= ±1 by the arithmetic nth root of the positive number a, it results that the theory of the equations of the form "1 is almost equivalent to that of equations with two coefficients not zero. For many purposes it is convenient to study equations from the standpoint of the number of coefficients which are supposed to differ from zero, especially when this number does not exceed 3, but in the present monograph the classification is made with respect to the degrees of the unknowns. The interesting properties which result from the assumption that the coefficients represent successively the various terms of sequences of numbers have been left untouched for want of space. |