Peter J. Cameron Introduction to Algebra Second edition, Oxford University Press, 2008 What is algebra? | Exercises | Other material | The first edition

• Hardback: 9780198569138
• Paperback: 9780198527930

You can see the preface and table of contents here.

This page will point you to a list of misprints, solutions to odd-numbered exercises, further exercises, additional material, etc.

What is algebra?

To criticize mathematics for its abstraction is to miss the point entirely. Abstraction is what makes mathematics work. If you concentrate too closely on too limited an application of a mathematical idea, you rob the mathematician of his most important tools: analogy, generality, and simplicity. Mathematics is the ultimate in technology transfer. Ian Stewart, Does God play dice? The mathematics of chaos, Penguin, London, 1990.

In other words, in algebra, we can prove a theorem in (say) ring theory and apply it to integers, polynomials, matrices, and so on. We don't need to do the work over again in each new situation we meet. Also, the more general theorem may be easier, since inessential detail is stripped away.

Here is Doctor Johnson's definition of algebra in his dictionary:

This is a peculiar kind of arithmetick, which takes the quantity sought, whether it be a number or a line, or any other quantity, as if it were granted, and by means of one or more quantities given, proceeds by consequence, till the quantity at first only supposed to be known, or at least some power thereof, is found to be equal to some quantity or quantities which are known, and consequently itself is known.

Read what other people have said about algebra here.

Exercises

Solutions

• Chapter 1
• Chapter 2
• Chapter 3
• Chapter 4
• Chapter 5
• Chapter 6
• Chapter 7 (part)
• Chapter 8

Further exercises

• Twenty-three problems on groups
• Nine problems on rings

Misprints

Lecture notes, etc.

• Here are course notes for four algebra courses I have taught recently:
  • Introduction to Algebra
  • Algebraic Structures
  • Linear Algebra
  • Classical Groups
• Here are web pages for my other textbooks:
  • Combinatorics: Topics, Techniques, Algorithms, published by Cambridge University Press;
  • Sets, Logic and Categories, published by Springer-Verlag.
  • Permutation Groups, published by Cambridge University Press.

Other resources

• Permutation groups resources at Queen Mary, University of London
• ATLAS of Finite Group Representations at Queen Mary, University of London
• Two systems for computational algebra:
  • GAP homepage at St Andrews University
  • MAGMA homepage at the University of Sydney
• From the MacTutor History of Mathematics Archive at St Andrews University:
  • History of Matrices and Determinants
  • History of Group Theory
• A Survey of Venn diagrams by Frank Ruskey and Mark Weston, from the Electronic Journal of Combinatorics
• Solving the quintic (worked example in Mathematica)
• Semigroups directory at the University of Southampton
• Nearrings homepage at the University of Linz
• Theory and Applications of Categories (electronic journal)

Peter J. Cameron
p.j.cameron(at)qmul.ac.uk
17 December 2007.