二次型算术PDF电子书下载

Chapter Ⅰ．The quadratic reciprocity law 1

1．Elementary facts 1

2．Structure of（Z/mZ）× 4

3．The quadratic reciprocity law 5

4．Lattices in a vector space 11

5．Modules over a principal ideal domain 12

Chapter Ⅱ．Arithmetic in an algebraic number field 15

6．Valuations and p-adic numbers 15

7．Hensel's lemma and its applications 22

8．Integral elements in algebraic extensions 25

9．Order functions in algebraic extensions 27

10．Ideal theory in an algebraic number field 35

Chapter Ⅲ．Various basic theorems 47

11．The tensor product of fields 47

12．Units and the class number of a number field 50

13．Ideals in an extension of a number field 57

14．The discriminant and different 59

15．Adeles and ideles 66

16．Galois extensions 71

17．Cyclotomic fields 75

Chapter Ⅳ．Algebras over a field 79

18．Semisimple and simple algebras 79

19．Central simple algebras 86

20．Quaternion algebras 95

21．Arithmetic of semisimple algebras 100

Chapter Ⅴ．Quadratic forms over a field 115

22．Algebraic theory of quadratic forms 115

23．Clifford algebras 120

24．Clifford groups and spin groups 127

25．Lower-dimensional cases 133

26．The Hilbert reciprocity law 140

27．The Hasse principle 143

Chapter Ⅵ．Deeper arithmetic of quadratic forms 153

28．Classification of quadratic forms over local and global fields 153

29．Lattices in a quadratic space 161

30．The genus and class of a lattice and a matrix 171

31．Integer-valued quadratic forms 179

32．Strong approximation in the indefinite case 186

33．Integer-valued symmetric forms 197

Chapter Ⅶ．Quadratic Diophantine equations 203

34．A historical perspective 203

35．Basic theorems of quadratic Diophantine equations 206

36．Classification of binary forms 213

37．New mass formulas 224

38．The theory of genera 228

References 233

Index 235