## Problems in Algebraic Number TheoryAsking how one does mathematical research is like asking how a composer creates a masterpiece. No one really knows. However, it is a recognized fact that problem solving plays an important role in training the mind of a researcher. It would not be an exaggeration to say that the ability to do mathematical research lies essentially asking "well-posed" questions. The approach taken by the authors in Problems in Algebraic Number Theory is based on the principle that questions focus and orient the mind. The book is a collection of about 500 problems in algebraic number theory, systematically arranged to reveal ideas and concepts in the evolution of the subject. While some problems are easy and straightforward, others are more difficult. For this new edition the authors added a chapter and revised several sections. The text is suitable for a first course in algebraic number theory with minimal supervision by the instructor. The exposition facilitates independent study, and students having taken a basic course in calculus, linear algebra, and abstract algebra will find these problems interesting and challenging. For the same reasons, it is ideal for non-specialists in acquiring a quick introduction to the subject. |

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### Contents

Elementary Number Theory | 1 |

12 Applications of Unique Factorization | 6 |

13 The ABC Conjecture | 7 |

14 Supplementary Problems | 8 |

Euclidean Rings | 11 |

22 Gaussian Integers | 15 |

24 Some Further Examples | 19 |

25 Supplementary Problems | 23 |

The Ideal Class Group | 67 |

62 Finiteness of the Ideal Class Group | 69 |

63 Diophantine Equations | 71 |

64 Exponents of Ideal Class Groups | 73 |

65 Supplementary Problems | 74 |

Quadratic Reciprocity | 79 |

72 Gauss Sums | 82 |

73 The Law of Quadratic Reciprocity | 84 |

Algebraic Numbers and Integers | 25 |

32 Liouvilles Theorem and Generalizations | 27 |

33 Algebraic Number Fields | 30 |

34 Supplementary Problems | 36 |

Integral Bases | 39 |

42 Existence of an Integral Basis | 41 |

43 Examples | 44 |

44 Ideals in OK | 47 |

45 Supplementary Problems | 48 |

Dedekind Domains | 51 |

52 Characterizing Dedekind Domains | 53 |

53 Fractional Ideals and Unique Factorization | 55 |

54 Dedekinds Theorem | 61 |

55 Factorization in Ok | 63 |

56 Supplementary Problems | 64 |

75 Primes in Special Progressions | 89 |

The Structure of Units | 3 |

83 Supplementary Problems | 19 |

Higher Reciprocity Laws | 21 |

92 Eisenstein Reciprocity | 26 |

93 Supplementary Problems | 29 |

Analytic Methods | 31 |

102 Zeta Functions of Quadratic Fields | 34 |

103 Dirichlets LFunctions | 37 |

104 Primes in Arithmetic Progressions | 38 |

Density Theorems | 43 |

112 Distribution of Prime Ideals | 50 |

113 The Chebotarev density theorem | 54 |

114 Supplementary Problems | 57 |

### Common terms and phrases

ABC Conjecture algebraic integer algebraic number field bounded Chapter class number coefficients congruent conjugate consider contains contradiction converges coprime Dedekind domain deduce define denote Dirichlet density discriminant divide equation Euclidean exists extension of algebraic Fermat's field of degree fractional ideal fundamental unit Galois Hence ideal class group ideal of Ok ideals of norm imaginary quadratic field implies induction infinitely many primes integral basis integral domain integral ideal integral solutions lattice point Lemma linear matrix minimal polynomial mod p2 mod q monic natural number nontrivial solution number of primes Number Theory odd prime positive integer previous exercise prime divisor prime ideals prime number primitive principal ideal domain product of prime Proof Prove quadratic reciprocity ramifies relatively prime result ring of integers root of unity satisfying set of primes squarefree Supplementary Problems Exercise Suppose transcendental unique factorization domain write

### Popular passages

Page v - What you have been obliged to discover by yourself leaves a path in your mind which you can use again when the need arises."4 This is an expressive way of saying that you have added to the accumulation of your mathematical intuition.

Page v - Precepts, I have thought fit to adjoin the Solutions of the following Problems.