1 Unique Factorisation in the Natural Numbers .................. 1
1.1 The Natural Numbers ..................................... 2
1.2 Euclid's Algorithm ...................................... 3
1.3 The Fundamental Theorem of Arithmetic ................... 8
1.4 The Gaussian Integers ................................... 9
1.5 Another Application of the Gaussian Integers ........... 14
2 Number Fields ............................................... 17
2.1 Algebraic Numbers ...................................... 17
2.2 Minimal Polynomials .................................... 20
2.3 The Field of Algebraic Numbers ......................... 21
2.4 Number Fields .......................................... 25
2.5 Integrality ............................................ 28
2.6 The Ring of All Algebraic Integers ..................... 31
2.7 Rings of Integers of Number Fields ..................... 35
3 Fields, Discriminants and Integral Bases .................... 39
3.1 Embeddings ............................................. 40
3.2 Norms and Traces ....................................... 44
3.3 The Discriminant ....................................... 47
3.4 Integral Bases ......................................... 51
3.5 Further Theory of the Discriminant ..................... 53
3.6 Rings of Integers in Some Cubic and Quartic Fields ..... 57
3.6.1 K =  (√2,√3) .................................... 57
3.6.2 К = (√-2,√-5) .................................. 59
3.6.3 K = (3√2) ...................................... 60
3.6.4 К = (3√175) .................................... 61
4 Ideals ...................................................... 65
4.1 Uniqueness of Factorisation Revisited .................. 66
4.2 Non-unique Factorisation in Quadratic Number Fields .... 67
4.3 Kummer's Ideal Numbers ................................. 71
4.4 Ideals ................................................. 73
4.5 Generating Sets for Ideals ............................. 76
4.6 Ideals in Quadratic Fields ............................. 79
4.7 Unique Factorisation Domains and Principal Ideal
Domains ................................................ 81
4.8 The Noetherian Property ................................ 84
5 Prime Ideals and Unique Factorisation ....................... 87
5.1 Some Ring Theory ....................................... 87
5.2 Maximal Ideals ......................................... 94
5.3 Prime Ideals ........................................... 96
5.4 Unique Factorisation into Prime Ideals ................. 99
5.5 Coprimality ........................................... 102
5.6 Norms of Ideals ....................................... 104
5.7 The Class Group ....................................... 105
5.8 Splitting of Primes ................................... 107
5.9 Primes in Quadratic Fields ............................ 112
6 Imaginary Quadratic Fields ................................. 113
6.1 Units ................................................. 113
6.1.1 d ≡ 2, 3 (mod 4) ............................... 114
6.1.2 d ≡ 1 (mod 4) .................................. 114
6.1.3 Summary ........................................ 115
6.2 Euclidean Imaginary Quadratic Fields .................. 116
6.2.1 d ≡ 2, 3 (mod 4) ............................... 116
6.2.2 d ≡ l (mod 4) .................................. 117
6.3 Quadratic Forms ....................................... 120
6.4 Reduction Theory ...................................... 123
6.5 Class Numbers and Quadratic Forms ..................... 131
6.5.1 d ≡ 2, 3 (mod 4) ............................... 134
6.5.2 d ≡ 1 (mod 4) .................................. 142
6.6 Counting Quadratic Forms .............................. 143
7 Lattices and Geometrical Methods ........................... 149
7.1 Lattices .............................................. 149
7.2 Geometry of Number Fields ............................. 153
7.3 Finiteness of the Class Number ........................ 158
7.4 Dirichlet's Unit Theorem .............................. 164
8 Other Fields of Small Degree ............................... 169
8.1 Continued Fractions ................................... 170
8.2 Continued Fractions of Square Roots ................... 176
8.3 Real Quadratic Fields ................................. 180
8.4 Biquadratic Fields .................................... 184
8.5 Cubic Fields .......................................... 188
9 Cyclotomic Fields and the Fermat Equation .................. 191
9.1 Definitions ........................................... 191
9.2 Discriminants and Integral Bases ...................... 195
9.3 Gauss Sums and Quadratic Reciprocity .................. 198
9.4 Remarks on Fermat's Last Theorem ...................... 202
10 Analytic Methods ........................................... 207
10.1 The Riemann Zeta Function ............................. 207
10.2 The Functional Equation of the Riemann Zeta Function .. 211
10.3 Zeta Functions of Number Fields ....................... 214
10.4 The Analytic Class Number Formula ..................... 215
10.5 Explicit Class Number Formulae ........................ 223
10.6 Other Embeddings ...................................... 226
11 The Number Field Sieve ..................................... 231
11.1 The RSA Cryptosystem and the Problem of
Factorisation ......................................... 231
11.2 The Quadratic Sieve ................................... 233
11.3 The Number Field Sieve: A First Example ............... 237
11.4 Index Calculus ........................................ 238
11.5 Prime Ideals and the Algebraic Factorbase ............. 241
11.6 Further Obstructions .................................. 244
11.7 The General Case ...................................... 248
11.8 Closing Comments ...................................... 253
Appendix A: Solutions and Hints to Exercises .................. 257
Index ......................................................... 289
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