代数数论讲义
2000-12
世界图书出版公司
Erich Hecke
242
11
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Hecke was certainly one of the masters, and in fact, the study of Hecke Lseries and Hecke operators has permanently embedded his name in the fabric of number theory. It is a rare occurrence when a master writes a basic book,and Hecke＇s Lectures on the Theory of Algebraic Numbers has become a classic. To quote another master, Andre Weil: "To improve upon Hecke, in a treatment along classical lines of the theory of algebraic numbers, would be a futile and impossible task."
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 14-15CHAPTER 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 34-40CHAPTER 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 nPrimitive 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 lawCHAPTER 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 56-59CHAPTER 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 theCHAPTER Ⅶ The Quadratic Number FieldCHAPTER Ⅷ The Law of Quadratic Reciprocity in Arbitrary Number FieldsChronological TableReferences
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E.Hecke是20世纪著名的数学家，他的这本《代数数论讲义》是一本很好的代数数论入门的书。