Introduction to Classical Mathematics I

6Et moi,., si j'avait su comment en revenir, One service mathematics has rendered the human mce. It has put common sense back je n'y serais point alle.' Jules Verne where it belongs, on the topmost shelf nCllt to the dusty canister labelled 'discarded non­ sense'. The series is divergent; therefore we may be able to do something with it. Eric T. Bell O. Heaviside Mathematics is a tool for thought. A highly necessary tool in a world where both feedback and non­ linearities abound. Similarly, all kinds of parts of mathematics serve as tools for other parts and for other sciences. Applying a simple rewriting rule to the quote on the right above one finds such statements as: 'One service topology has rendered mathematical physics.'; 'One service logic has rendered com­ puter science.'; 'One service category theory has rendered mathematics.'. All arguably true. And all statements obtainable this way form part of the raison d'etre of this series.

Inhaltsangabe1. 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.