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