### II.3 The Development of Abstract Algebra

*Karen Hunger Parshall*

### 1 Introduction

What is algebra? To the high school student encountering it for the first time, algebra is an unfamiliar abstract language of *x*’s and *y*’s, *a*’s and *b*’s, together with rules for manipulating them. These letters, some of them variables and some constants, can be used for many purposes. For example, one can use them to express straight lines as equations of the form *y* = *ax* + *b*, which can be graphed and thereby visualized in the Cartesian plane. Furthermore, by manipulating and interpreting these equations, it is possible to determine such things as what a given line’s root is (if it has one)—that is, where it crosses the *x*-axis—and what its slope is—that is, how steep ...