Algebra

A book in 22 chapters, from "think of a number" to the reason no formula solves every equation of degree five, and on to the algebra mathematicians use today. Each chapter starts from a question, explains every new word, proves what it claims, works examples step by step, and ends with questions whose answers are folded underneath.

How to read it

• In order. Each chapter needs only the ones before it; the Next link at the end of every chapter walks you through.
• Levels inside a chapter. The box at the top of each chapter says which sections are beginner, intermediate, advanced or expert. On a first read, skip the harder sections and come back later.
• Slower help. The first chapters point to the matching lessons on the Topics shelf, which teach the same ideas more gently, with pictures, for younger readers.
• Do the questions. Try each one before opening its answer.

Contents

Part	Chapter	Level
I. The language	1. The language of algebra	beginner
	2. Equations and inequalities	beginner
	3. Lines and linear systems	beginner to intermediate
II. Polynomials and functions	4. Polynomials and factoring	intermediate
	5. Quadratics	intermediate
	6. Functions	intermediate
	7. Exponents and logarithms	intermediate
	8. Sequences, series and induction	intermediate
	9. Complex numbers	intermediate to advanced
III. The theory of equations	10. Roots and coefficients	advanced
	11. Cubics and quartics	advanced
	12. The fundamental theorem of algebra	advanced
IV. Abstract algebra	13. Groups	advanced
	14. Subgroups, cosets and Lagrange	advanced
	15. Homomorphisms and quotients	advanced
	16. Group actions and Sylow	advanced to expert
	17. Rings and ideals	advanced
V. Fields, Galois and the frontier	18. Factorization in rings	advanced to expert
	19. Fields and extensions	expert
	20. Galois theory	expert
	21. Modules and representations	expert
	22. Where algebra goes	expert

The glossary defines every term, with a link to the chapter that teaches it.

Three ways in

• Starting out, or coming back after years away: read chapters 1 to 9. That is all of school algebra, with the reasons written out.
• You know school algebra and want to see why it works: start at chapter 9 or 10, then go straight through to chapter 20, where the question of chapter 11 (why is there no formula for degree five?) is finally answered.
• You have met groups before: start at chapter 17 and use the glossary to look back.

Linear algebra (vectors, matrices, eigenvalues) gets its own book on this shelf; this one touches it only where groups and modules need it.