I. The Foundations of Arithmetic in the Rational Number Field.- 1. Prime Decomposition.- 2. Divisibility.- 3. Congruences.- 4. The Structure of the Residue Class Ring mod m and of the Reduced Residue Class Group mod m.- 5. Quadratic Residues.- II. The Theory of Valued Fields.- 6. The Fundamental Concepts Regarding Valuations.- 7. Arithmetic in a Discrete Valued Field.- 8. The Completion of a Valued Field.- 9. The Completion of a Discrete Valued Field. The p-adic Number Fields.- 10. The Isomorphism Types of Complete Discrete Valued Fields with Perfect Residue Class Field.- 11. Prolongation of a Discrete Valuation to a Purely Transcendental Extension.- 12. Prolongation of the Valuation of a Complete Field to a Finite-Algebraic Extension.- 13. The Isomorphism Types of Complete Archimedean Valued Fields.- 14. The Structure of a Finite-Algebraic Extension of a Complete Discrete Valued Field.- 15. The Structure of the Multiplicative Group of a Complete Discrete Valued Field with Perfect Residue Class Field of Prime Characteristic.- 16. The Tamely Ramified Extension Types of a Complete Discrete Valued Field with Finite Residue Class Field of Characteristic p.- 17. The Exponential Function, the Logarithm, and Powers in a Complete Non-Archimedean Valued Field of Characteristic 0.- 18. Prolongation of the Valuation of a Non-Complete Field to a Finite-Algebraic Extension.- III. The Foundations of Arithmetic in Algebraic Number Fields.- 19. Relations Between the Complete System of Valuations and the Arithmetic of the Rational Number Field.- 20. Prolongation of the Complete System of Valuations to a Finite-Algebraic Extension.- 21. The Prime Spots of an Algebraic Number Field and their Completions.- 22. Decomposition into Prime Divisors, Integrality, and Divisibility.- 23. Congruences.- 24. The Multiples of a Divisor.- 25. Differents and Discriminants.- 26. Quadratic Number Fields.- 27. Cyclotomic Fields.- 28. Units.- 29. The Class Number.- 30. Approximation Theorems and Estimates of the Discriminant.- Index of Names.
