Algebra is the lingua franca of the mathematical sciences and the purpose of this little book is to explain what it is about. The laws governing algebra emerge from the behaviour of numbers, and one of our themes is how many of the rules and practices of arithmetic and algebra are consequences of a small collection of fundamental laws that represent familiar properties of ordinary integers.
The first half of the book establishes much of the algebra that has been a staple of secondary school mathematics for generations, which is based on finding unknowns in linear and quadratic equations, and this the reader meets in the first four chapters. Modern algebra was born out of the struggle to solve equations of degree higher than 2, and the first part of the book culminates in Chapter 5 with finding solutions of general cubic equations, the roots of which are not necessarily just simple fractions but may involve so-called irrational and complex numbers.
The second half of the book introduces modern aspects of the subject and we look at algebra that is not based on the general behaviour of numbers but involves other kinds of mathematical objects. The topic of Chapter 6 is the arithmetic of remainders, which furnishes examples of a fundamental algebra type with two operations, namely that of a ring. Matrices are the central feature of Chapters 7, 8, and 9. The origin of matrices may be traced back thousands of years to ancient China but the topic only gained traction in the middle of the 19th century, from which point it has grown to become the primary vehicle for calculation throughout mathematics, physics, and the social sciences. The historical significance of matrix theory in pure mathematics, however, is that it provided an important example of another type of algebra apart from number fields. The final chapter introduces vector spaces and finite fields.
Many aspects of algebra are touched upon in the course of the book and every piece matters. My hope is that readers will see the parts of the jigsaw come together as they move through the book and thereby gain an appreciation of algebra as a whole. Modern abstract algebra is firmly based on what are known as groups, rings, fields, and vector spaces. The reader is made aware of these constructs through examples that emerge in the development of the text. Only after that are these ideas introduced in a more formal fashion. The intention is that the reader will be left with both an overview of elementary mathematics and a taste for and an insight into contemporary aspects of the vast world of algebra.
Peter M. Higgins,
Colchester, 2015