Algebraic Number Theory J.S. Milne Version 3.01 September 28, 2008 An algebraic number field is a finite extension of Q; an algebraic number is an element of an algebraic number field. Algebraic number theory studies the arithmetic of algebraic number fields — the ring of integers in the number field, the ideals and units in the ring of integers, the extent to which unique factorization holds, and so on. An abelian extension of a field is a Galois extension of the field with abelian Galois group. Class field theory describes the abelian extensions of a number field in terms of the arithmetic of the field. These notes are concerned with algebraic number theory, and the sequel with class field theory. The original version was distributed during the teaching of a second-year graduate course. BibTeX information @misc{milneANT, author={Milne, James S.}, title={Algebraic Number Theory (v3.01)}, year={2008}, note={Available at www.jmilne.org/math/}, pages={155+viii} } v2.01 (August 14, 1996). First version on the web. v2.10 (August 31, 1998). Fixed many minor errors; added exercises and an index; 138 pages. v3.00 (February 11, 2008). Corrected; revisions and additions; 163 pages. v3.01 (September 28, 2008). Fixed problem with hyperlinks; 163 pages. Available at www.jmilne.org/math/ Please send comments and corrections to me at the address on my web page. The photograph is of the Fork Hut, Huxley Valley, New Zealand. Copyright c 1996, 1998, 2008, J.S. Milne. Single paper copies for noncommercial personal use may be made without explicit permis- sion from the copyright holder. Contents Contents 2 Notations. 5 Prerequisites . 5 References . 5 Acknowledgements . 5 Introduction . 1 Exercises . 6 1 Preliminaries from Commutative Algebra 7 Basic definitions . 7 Ideals in products of rings . 8 Noetherian rings . 8 Noetherian modules . 9 Local rings . 10 Rings of fractions . 11 The Chinese remainder theorem . 12 Review of tensor products . 14 Exercise . 17 2 Rings of Integers 18 First proof that the integral elements form a ring . 18 Dedekind’s proof that the integral elements form a ring . 19 Integral elements . 21 Review of bases of A-modules . 24 Review of norms and traces . 24 Review of bilinear forms . 25 Discriminants . 26 Rings of integers are finitely generated . 28 Finding the ring of integers . 30 Algorithms for finding the ring of integers . 33 Exercises . 37 3 Dedekind Domains; Factorization 39 Discrete valuation rings . 39 Dedekind domains . 41 Unique factorization of ideals . 41 The ideal class group . 45 2 Discrete valuations . 48 Integral closures of Dedekind domains . 49 Modules over Dedekind domains (sketch). 50 Factorization in extensions . 51 The primes that ramify . 52 Finding factorizations . 55 Examples of factorizations . 56 Eisenstein extensions . 58 Exercises . 59 4 The Finiteness of the Class Number 61 Norms of ideals . 61 Statement of the main theorem and its consequences . 63 Lattices . 66 Some calculus . 70 Finiteness of the class number . 73 Binary quadratic forms . 74 Exercises . 76 5 The Unit Theorem 78 Statement of the theorem . 78 Proof that UK is finitely generated . 80 Computation of the rank . 81 S-units . 83 Example: CM fields . 83 Example: real quadratic fields . 84 Example: cubic fields with negative discriminant . 85 Finding .K/ .................................... 87 Finding a system of fundamental units . 87 Regulators . 87 Exercises . 88 6 Cyclotomic Extensions; Fermat’s Last Theorem. 89 The basic results . 89 Class numbers of cyclotomic fields . 95 Units in cyclotomic fields . 95 The first case of Fermat’s last theorem for regular primes . 96 Exercises . 98 7 Valuations; Local Fields 99 Valuations . 99 Nonarchimedean valuations . 100 Equivalent valuations . 101 Properties of discrete valuations . 103 Complete list of valuations for the rational numbers . 103 The primes of a number field . 105 The weak approximation theorem . 107 Completions . 108 Completions in the nonarchimedean case . 109 Newton’s lemma . 113 Extensions of nonarchimedean valuations . 116 Newton’s polygon . 118 Locally compact fields . 119 Unramified extensions of a local field . 120 Totally ramified extensions of K . 122 Ramification groups . 123 Krasner’s lemma and applications . 124 Exercises . 126 8 Global Fields 128 Extending valuations . 128 The product formula . 130 Decomposition groups . 132 The Frobenius element . 134 Examples . 135 Computing Galois groups (the hard way) . 137 Computing Galois groups (the easy way) . 137 Applications of the Chebotarev density theorem . 142 Exercises . 144 A Solutions to the Exercises 145 B Two-hour examination 151 Bibliography 152 Index 154 Notations. We use the standard (Bourbaki) notations: N 0; 1; 2; : : : ; Z ring of integers; R D f g D D field of real numbers; C field of complex numbers; Fp Z=pZ field with p elements, D D D p a prime number. For integers m and n, m n means that m divides n, i.e., n mZ. Throughout the notes, j 2 p is a prime number, i.e., p 2; 3; 5; : : :. D Given an equivalence relation, Œ denotes the equivalence class containing . The empty set is denoted by . The cardinality of a set S is denoted by S (so S is the number ; j j j j of elements in S when S is finite). Let I and A be sets; a family of elements of A indexed by I , denoted .ai /i I , is a function i ai I A. X YX is a subset2 of Y (not necessarily7! W proper);! X def YX is defined to be Y , or equals Y by definition; D X YX is isomorphic to Y ; X YX and Y are canonically isomorphic (or there is a given or unique isomorphism); ' , denotes an injective map; ! denotes a surjective map. It is standard to use Gothic (fraktur) letters for ideals: abcmnpqABCMNPQ abcmnpqABCMNPQ Prerequisites The algebra usually covered in a first-year graduate course, for.