Mathematics and Its Applications

1. Congruences.- 2. Quadratic forms.- 3. Division of the circle (cyclotomy).- 4. Theory of surfaces.- 5. Harmonic analysis.- 6. Prime numbers in arithmetic progressions.- 7. Theory of algebraic equations.- 8. The beginnings of complex function theory.- 9. Entire functions.- 10. Riemann surfaces.- 11. Meromorphic differentials on closed Riemann surfaces.- 12. The theorems of Abel and Jacobi.- 13. Elliptic functions.- 14. Riemannian geometry.- 15. On the number of primes less than a given magnitude.- 16. The origins of algebraic number theory.- 17. Field theory.- 18. Dedekind's theory of ideals.- 19. The ideal class group and the group of units.- 20. The Dedekind ?-function.- 21. Quadratic forms and quadratic fields.- 22. The different and the discriminant.- 23. Theory of algebraic functions of one variable.- 24. The geometry of numbers.- 25. Normal extensions of algebraic number- and function fields.- 26. Entire functions with growth of finite order.- 27. Proof of the prime number theorem.- 28. Combinatorial topology.- 29. The idea of a Riemann surface.- 30. Uniformisation.- Appendix 1. Rings.- A1.1 Basic ring concepts.- A1.2 Euclidean rings.- A1.3 The characteristic of a ring.- A1.4 Modules over euclidean rings.- Al.5 Construction of fields.- A1.6 Polynomials over fields.- Appendix 2. Set theoretic topology.- A2.1 Definition of a topological space.- A2.2 Compact spaces.- Appendix 3. Green's theorem.- Appendix 4. Euclidean vector and point spaces.- Appendix 5. Projective spaces.- Name index.- General index.