| CHAPTER Ⅰ Elements of Rational Number Theory 1. Divisibility, Greatest Common Divisors, Modules, Prime Numbers, and the Fundamental Theorem of Number Theory Theorems 1-5 2. Congruences and Residue Classes Euler''sfunction （n）.Ferrnat'' s theorem. Theorems 6-9 3. Integral Polynomials, Functional Congruences, and Divisibility mod p Theorems lO-13a 4. Congruences of the First Degree Theorems 14-15 CHAPTER Ⅱ Abelian Groups 5. The General Group Concept and Calculation with Elements of a Group Theorems 16-18 6. Subgroups and Division of a Group by a Subgroup Order of elements. Theorems 19-21 7. Abelian Groups and the Product of Two Abeliun Groups Theorems 22-25 8. Basis of an Abelian Group The basis number ora group belonging to a prime number. Cyclic groups. Theorems 26-28 9. Composition of Cosets and the Factor Group Theorem 29 10. Characters of Abelian Groups The group of characters. Determination of all subgroups. Theorems 30-33 11. Infinite Abelian Groups Finite basis of such a group and basis for a subgroup. Theorems 34-40 CHAPTER Ⅲ Abelian Groups in Rational Number Theory 12. Groups of Integers under Addition and Multiplication Theorem 41 13. Structure of the Group R n of the Residue Classes mod n Relatively Prime to n Primitive numbers mod p and mod p2. Theorems 42-45 14. Power Residues Binomial congruences. Theorems 46-47 15. Residue Characters of Numbers mod n 16. Quadratic Residue Characters mod n On the quadratic reciprocity law CHAPTER Ⅳ Algebra of Number Fields 17. Number Fields, Polynomials over Number Fields, and Irreducibility Theorems 48-49 18. Algebraic Numbers over k Theorems 50-519 19. Algebraic Number Fields over k Simultaneous ad unction of several numbers. The conjugate numbers. Theorems 52-55 20. Generating Field Elements, Fundamental Systems, and Subfields of K0 Theorems 56-59 CHAPTER V General Arithmetic of Algebraic Number Fields 21. Definition of Algebraic Integers, Divisibility, and Units Theorems 60-63 22. The Integers of a Field as an Abelian Group: Basis and Discriminant of the Field Moduli. Theorem 64 23. Factorization of Integers in K: Greatest Common Divisors which Do Not Belong to the Field 24. Definition and Basic Properties of Ideals Product of ideals. …… CHAPTER VI Introduction of Transcendental Methods into the CHAPTER Ⅶ The Quadratic Number Field CHAPTER Ⅷ The Law of Quadratic Reciprocity in Arbitrary Number FieldsChronological Table References |
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