Number Theory

Part I. The Foundations of Arithmetic in the Rational Number Field 1
Chapter 1. Prime Decomposition 2
Function Fields 7
Chapter 2. Divisibility 8
Function Fields 22
Chapter 3. Congruences 24
Function Fields 38
The Theory of Finite Fields 40
Chapter 4. The Structure of the Residue Class Ring mod m and of the Reduced
Residue Class Group mod m 42
1. General Facts Concerning Direct Products and Direct Sums 42
2. Direct Decomposition of the Residue Class Ring mod m and of the Reduced Residue Class Group mod m 46
3. The Structure of the Additive Group of the Residue Class Ring mod m 65
4. On the Structure of the Residue Class Ring mod p^mu 56
5. The Structure of the Reduced Residue Class Group mod p^mu 57
Function Fields 63
Chapter 5. Quadratic Residues 64
1. Theory of the Characters of a Finite Abelian Group 64
2. Residue Class Characters and Numerical Characters mod m 69
3. The Basic Facts Concerning Quadratic Residues 73
4. The Quadratic Reciprocity Law for the Legendre Symbol 77
5. The Quadratic Reciprocity Law for the Jacobi Symbol 8 3
6. The Quadratic Reciprocity Law 8s Product Formula for the Hilbert Symbol 92
7. Special Cases of Dirichlet's Theorem on Prime Numbers in Reduced Residue Classes 96
Function Field 100
Part II. The Theory of Valued Fields 106
Chapter 6. The Fundamental Conceptn Regarding Valuations 105
1. The Definition of a Valuation; Equivalent Valuations 106
2. Approximation Independence and Multiplicative Independence of Valuations 109
3. Valuations of the Prime Field 113
4. Value Groups and Residue Class Fields 122
Function Fields 126
Chapter 7. Arithmetic in a Discrete Valued Field 129
Divisors from an Ideal-Theoretic Standpoint 133
Chapter 8. The Completion of a Valued Field. 136
Chapter 9. The Completion of a Discrete Valued Field. The p-adic Number
Fields. 144
Function Fields 149
Chapter IO. The Isomorphism Types of Complete Discrete Valued Fields with Perfect Residue Class Field 151
1. The Multiplicative Residue System in the Case of Prime Characteristic
 162
2. The Equal-Characteristic Case with Prime Characteristic 154
3. The Multiplicative Residue System in the p-adic Number Field 155
4. Witt's Vector Calculus 166
5. Construction of the General p-adic Field 161
6. The Unequal-Characteristic Case 165
7. Isomorphic Residue Systems in the Case of Characteristic 0 169
8. The Isomorphic Residue Systems for a Rational Function Field 174
9. The Equal-Characteristic Case with Characteristic 0 174
Chapter 11. Prolongation of a Discrete Valuation to a Purely Transcendental
Extension 176
Chapter 12. Prolongation of the Valuation of a Complete Field to a Finite-Algebraic
Extension 182
1. The Proof of Existence 184
2. The Proof of Completeness 188
3. The Proof of Uniqueness 190
Chapter 13. The Isomorphism Types of Complete Archimedean Valued Fields 191
Chapter 14. The Structure of a Finite-Algebraic Extension of a Complete Discrete Valued Field 194
1. Embedding of the Arithmetic 195
2. The Totally Ramified Case 201
3. The Unramified Case with Perfect Residue Class Field 203
4. The General Case with Perfect Residue Class Field 208


5. The General Case with Finite Residue Class Field 210
Chapter 16. The Structure of the Multiplicative Group of a Complete Discrete
Valued Field with Perfect Residue Class Field of Prime Characteristic 213
1. Reduction to the One-Unit Group and its Fundamental Chain of Subgroups 213
2. The One-Unit Group as an Abelian Operator Group 215
3. The Field of nth Roots of Unity over a p-adic Number Field 219
4. The Structure of the One-Unit Group in the Equal-Characteristic
Case with Finite Residue Class Field 226
5. The Structure of the One-Unit Group in the p-adic Case 228
6. Construction of a System of Fundamental One-Units in the Q-adic Case 233
7. The One-Unit Group for Special p-adic Number Fields 246
8. Comparison of the Basis Representation of the Multiplicative Group in the p-adic Case and the Archimedean Case 247
Chapter 16. The Tamely Ramified Extension Types of a Complete Discrete Valued Field with Finite Residue Class Field of Characteristic p 248
Chapter 17. The Exponential Function, the Logarithm, and Powers in a Complete
Non-Archimedean Valued Field of Characteristic 0 255
1. Integral Power Series in One Indeterminate over an Arbitrary Field 255
2. Integral Power Series in One Variable in a Complete Non-Archimedean
Valued Field 256
3. Convergence 262
4. Functional Equations and Mutual Relations 266
5. The Discrete Case 274
6. The Equal-Characteristic Case with Characteristic 0 276
Chapter 18. Prolongation of the Valu&ion of a Non-Complete Field to a Finite-Algebraic Extension 276
1. Representations of a Separable Finite-Algebraic Extension over an Arbitrary Extension of the Ground Field 279
2. The Ring Extension of a Separable Finite-Algebraic Extension by an Arbitrary Ground Field Extension, or the Tensor Product of the Two Field Extensions 283
3. The Characteristic Polynomial 294
4. Supplements for Inseparable Extensions 297
5. Prolongation of a Valuation 298
6. The Discrete Case 302
7. The Archimedean Case I 305
Part III. The Foundations of Arithmetic in Algebraic Number Fields 309
Chapter 19. Relations Between the Complete System of Valuations and the Arithmetic of the Rational Number Field 310
1. Finiteness Properties 310
2. Characterizations in Divisibility Theory 310
3. The Product Formula for Valuations 311
4. The Sum Formula for the Principal Parts 312
Function Fields 315
The Automorphisms of a Rational Function Field 322
Chapter 20. Prolongation of the Complete System of Valuations to a Finite-Algebraic Extension 324
Function Fields 328
Concluding Remarks 334
Chapter 21. The Prime Spots of an Algebraic Number Field and their Completions 335
Function Fields 337
Chapter 22. Decomposition into Prime Divisors, Integrality, and Divisibility 338
1. The Canonical Homomorphism of the Multiplicative Group into the Divisor Group 338
2. Embedding of Divisibility Theory under Finite-Algebraic Extension 344
3. Algebraic Characterization of Integral Algebraic Numbers 361
4. Quotient Representation 363
Function Fields 354
Constant Fields, Constant Extensions 362
Chapter 23. Congruences 367
1. Ordinary Congruence 368
2. Multiplicative Congruence 370 
Function Fields 372
Chapter 24. The Multiples of a Divisor 372
1. Field Bases 373
2. The Ideal Property, Ideal Bases 375
3. Congruences for Integral Elements 378
4. Divisors from the Ideal-Theoretic Standpoint 383
5. Further Remarks Concerning Divisors and Ideals .389


Function Fields 393
Constant Fields for p. Characterization of Prime Divisors by Homomorphisms. Decomposition Law under an Algebraic Constant Extension 402
The Rank of the Module of Multiples of a Divisor 419
Chapter 25. Differents and Discriminants 429
1. Composition Formula for the Trace and Norm. The Divisor Trace 430
2. Definition of the Different and Discriminant 433
3. Theorems on Differents and Discriminants in the Small 435
4. The Relationship Between Differents and Discriminants in the Small and in the Large 442
5. Theorems on Differents and Discriminants in the Large 443
6. Common Inessential Discriminant Divisors 452
7. Examples 457
Function Fields 462
The Number of First-Degree Prime Divisors in the Case of a 

Finite Constant Field 464
Differentials 467
The Riemann-Roth Theorem and its Consequences 473
Disclosed Algebraic Function Fields 484
Chapter 26. Quadratic Number Fields 494
1. Generation in the Large and in the Small 494
2. The Decomposition Law. 496
3. Discriminants, Integral Bases 498
4. Quadratic Residue Characters of the Discriminant of an Arbitrary Algebraic Number Field 501
5. The Quadratic Number Fields as Class Fields 504
6. The Hilbert Symbol as Norm Symbol 505
7. The Norm Theorem 510
8. A Necessary Condition for Principal Divisors. Genera 512
Chapter 27. Cyolotomic Fields 520
1. Generation 520
2. The Decomposition Law 522
3. Discriminants, Integral Bases 523
4. The Quadratic Number Fields as Subfields of Cyclotomic Fields 528
Chapter 28. Units 536
I. Preliminaries 536
2. Proofs 540
3. Extension 549
4. Examples and Applications 550
Chapter 29. The Class Number 562
1. Finiteness of the Class Number 562
2. Consequences 563
3. Examples and Applications 565
Function Fields 594
Chapter 30. Approximation Theorema and Estimates of the Discriminant 696
1. The Most General Requirements on Approximating Zero 597
2. Minkowski's Lattice-Point Theorem 600
3. Application to Convex Bodies within the Norm-one Hypersurface 603
4. Consequences of the Discriminant Estimate 616
Function Fields 626
Indexof Names 628
Subject Index 629