This textbook provides an accessible account of the history of abstract algebra, tracing a range of topics in modern algebra and number theory back to their modest presence in the seventeenth and eighteenth centuries, and exploring the impact of ideas on the development of the subject.

Introduction.- 1 Simple quadratic forms.- 2 Fermat's Last Theorem.- 3 Lagrange's theory of quadratic forms.- 4 Gauss's Disquisitiones Arithmeticae.- 5 Cyclotomy.- 6 Two of Gauss's proofs of quadratic reciprocity.- 7 Dirichlet's Lectures.- 8 Is the quintic unsolvable?.- 9 The unsolvability of the quintic.- 10 Galois's theory.- 11 After Galois - Introduction.- 12 Revision and first assignment.- 13 Jordan's Traité.- 14 Jordan and Klein.- 15 What is 'Galois theory'?.- 16 Algebraic number theory: cyclotomy.- 17 Dedekind's first theory of ideals.- 18 Dedekind's later theory of ideals.- 19 Quadratic forms and ideals.- 20 Kronecker's algebraic number theory.- 21 Revision and second assignment.- 22 Algebra at the end of the 19th century.- 23 The concept of an abstract field.- 24 Ideal theory.- 25 Invariant theory.- 26 Hilbert's Zahlbericht.- 27 The rise of modern algebra - group theory.- 28 Emmy Noether.- 29 From Weber to van der Waerden.- 30 Revision and final assignment.- A Polynomial equations in the 18th Century.- B Gauss and composition of forms.- C Gauss on quadratic reciprocity.- D From Jordan's Traité.- E Klein's Erlanger Programm.- F From Dedekind's 11th supplement.- G Subgroups of S4 and S5.- H Curves.- I Resultants.- Bibliography.- Index.