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ОбложкаJarvis F. Algebraic number theory. - Cham: Springer, 2014. - xiii, 292 p. - (Springer undergraduate mathematics series). - Bibliogr.: p. 287. - Ind.: p.289-292. - ISBN 978-3-319-07544-0; ISSN 1615-2085
Шифр: (И/В14-J25) 02
 

Место хранения: 02 | Отделение ГПНТБ СО РАН | Новосибирск

Оглавление / Contents
 
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 = fig.1(√2,√3) .................................... 57
        3.6.2  К = fig.1(√-2,√-5) .................................. 59
        3.6.3  K = fig.1(3√2) ...................................... 60
        3.6.4  К = fig.1(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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