Theory of Algebraic Integers

Front Cover......Page 1
Title Page......Page 4
Copyright and ISBN......Page 5
Contents......Page 6
 Part-I: Translator's introduction......Page 10
0.1 General remarks......Page 12
0.2.1 Pythagorean triples......Page 15
0.2.2 Divisors and prime factorisation......Page 16
0.2..4 Diophantus......Page 17
0.3.1 Fermat......Page 19
0.3.2 The grit in the oyster......Page 21
0.3.3 Reduction of forms......Page 22
0.3.4 Lagrange's proof of the two squares theorem......Page 24
0.3.5 Primitive roots and quadratic residues......Page 25
0.3.6 Composition of forms......Page 26
0.3.7 The class group......Page 28
0.4.1 The need for generalised "integers"......Page 30
0.4.2 Gaussian integers......Page 31
0.4.3 Gaussian primes......Page 33
0.4.4 Imaginary quadratic integers......Page 34
0.4.5 The failure of unique prime factorisation......Page 36
0.5.1 Fermat's last theorem......Page 38
0.5.2 The cyclotomic integers......Page 39
0.5.3 Cyclotomic integers and quadratic integers......Page 41
0.5.4 Quadratic reciprocity......Page 45
0.5.5 Other reciprocity laws......Page 47
0.6.1 Definition......Page 48
0.6.2 Basic properties......Page 49
0.6.3 Class numbers......Page 50
0.6..4 Ideal numbers and ideals......Page 51
0.7.1 How the memoir came to be written......Page 53
0.7.2 Later development of ideal theory......Page 54
Acknowledgements......Page 56
Bibliography......Page 57
Part-II: Theory of algebraic integers......Page 60
Introduction......Page 62
§1. Modules and their divisibility......Page 71
§2. Congruences and classes of numbers......Page 73
§3. Finitely generated modules......Page 76
§4. Irreducible systems......Page 80
§ 5. The rational integers......Page 92
§6. The complex integers of Gauss......Page 93
§7. The domain o of numbers x + y qrt{-5}......Page 95
§8. Role of the number 2 in the domain o......Page 98
§9. Role of the numbers 3 and 7 in the domain o......Page 100
§10. Laws of divisibility in the domain o......Page 102
§11. Ideals in the domain o......Page 104
§12. Divisibility and multiplication of ideals in o......Page 107
§13. The domain of all algebraic integers......Page 112
§ 14. Divisibility of integers......Page 114
§ 15. Fields of finite degree......Page 115
§ 16. Conjugate fields......Page 117
§ 17. Norms and discriminants......Page 120
§ 18. The integers in a field \Sigma of finite degree......Page 122
§19. Ideals and their divisibility......Page 128
§20. Norms......Page 130
§21. Prime ideals......Page 132
§22. Multiplication of ideals......Page 134
§23. The difficulty in the theory......Page 135
§24. Auxiliary propositions......Page 137
§25. Laws of divisibility......Page 138
§26. Congruences......Page 143
§27. Examples borrowed from circle division......Page 147
§28. Classes of ideals......Page 155
§29. The number of classes of ideals......Page 156
§30. Conclusion......Page 158
Index......Page 162