Preface Notation and terminology Chapter Ⅰ. The quadratic reciprocity law 1. Elementary facts 2. Structure of (Z/mZ)X 3. The quadratic reciprocity law 4. Lattices in a vector space 5. Modules over a principal ideal domain Chapter Ⅱ. Arithmetic in an algebrai uber field 6. Valuations and p-adi ubers 7. Hensel's lemma and its applications 8. Integral elements in algebraic extensions 9. Order functions in algebraic extensions 10. Ideal theory in an algebrai uber field Chapter Ⅲ. Various basic theorems 11. The tensor product of fields 12. Units and the class number of a number field 13. Ideals in an extension of a number field 14. The discriminant and different 15. Adeles and ideles 16. Galois extensions 17. Cyclotomic fields Chapter Ⅳ. Algebras over a field 18. Semisimple and simple algebras 19. Central simple algebras 20. ternion algebras 21. Arithmetic of semisimple algebras Chapter Ⅴ. dratic forms over a field 22. Algebraic theory of quadratic forms . Clifford algebras 24. Clifford groups and spin groups 25. Lower-dimensional cases 26. The Hilbert reciprocity law 27. The Hasse principle Chapter Ⅵ. Deeper arithmetic of quadratic forms 28. Classification of quadratic forms over local and global fields 29. Lattices in a quadratic space 30. The genus and class of a lattice and a matrix 31. Integer-valued quadratic forms 32. Strong approximation in the indefinite case 33. Integer-valued symmetric forms Chapter Ⅶ. dratic Diophantine equations 34. A historical perspective 35. Basic theorems of quadratic Diophantine equations 36. Classification of binary forms 37. New mass formulas 38. The theory of genera References Index