CLASS FIELD THEORY J.S. Milne Preface. These12 are the notes for Math 776, University of Michigan, Winter 1997, slightly revised from those handed out during the course. They have been substantially revised and expanded from an earlier version, based on my notes from 1993 (v2.01). My approach to class field theory in these notes is eclectic. Although it is possible to prove the main theorems in class field theory using neither analysis nor cohomology, there are major theorems that can not even be stated without using one or the other, for example, theorems on densities of primes, or theorems about the cohomology groups associated with number fields. When it sheds additional light, I have not hesitated to include more than one proof of a result. The heart of the course is the odd numbered chapters. Chapter II, which is on the cohomology of groups, is basic for the rest of the course, but Chapters IV, VI, and VIII are not essential for reading Chapters III, V, and VII. Except for its first section, Chapter I can be skipped by those not interested in explicit local class field theory. References of the form Math xxx are to course notes available at http://www.math.lsa.umich.edu/∼jmilne. Please send comments and corrections to me at [email protected]. Books including class field theory Artin, E., Algebraic Numbers and Algebraic Functions, NYU, 1951.(Reprintedby Gordon and Breach, 1967). Artin, E., and Tate, J., Class Field Theory. Notes of a Seminar at Princeton, 1951/52. (Harvard University, Mathematics Department, 1961; Benjamin, 1968; Addison Wesley, 1991). Cassels, J.W.S., and Fr¨ohlich, A., (Eds), Algebraic Number Theory, Academic Press, 1967. Fesenko, I.B., and Vostokov, S.V., Local Fields and their Extensions, AMS, 1993. Goldstein, L.J., Analytic Number Theory, Prentice Hall, 1971. Iwasawa, K., Local Class Field Theory, Oxford, 1986. Iyanaga, S., (ed.), The Theory of Numbers, North-Holland, 1975. Janusz, G.J., Algebraic Number Fields, Academic Press, 1973; Second Edition, AMS, 1996. Koch, H., Number Theory II, Algebraic Number Theory, Springer, 1992.(Ency- clopaedia of Mathematical Sciences, Vol. 62, Parshin, A.N., and Shafarevich, I.R., Eds). Lang, S., Algebraic Number Theory, Addison-Wesley, 1970. Neukirch, J., Class Field Theory, Springer, 1985. Neukirch, J., Algebraische Zahlentheorie, Springer, 1992. Serre, J-P., Corps Locaux, Hermann, 1962. 1May 6, 1997; v3.1 2Copyright 1996, 1997, J.S. Milne. You may make one copy of these notes for your own personal use. ii iii Weil, A., Basic Number Theory, Springer, 1967. The articles of Serre and Tate in Cassels and Fr¨ohlich 1967, and Janusz 1973, have been especially useful in the writing of these notes. Books including an introduction to class field theory Cohn, H., A Classical Invitation to Algebraic Numbers and Class Fields, Springer, 1978. Cox, D.A., Primes of the Form x2 + ny2: Fermat, Class Field Theory, and Complex Multiplication, Wiley, 1989. Marcus, D. A., Number Fields, Springer 1977. Sources for the history of algebraic number theory and class field theory Edwards, H.M., Fermat’s Last Theorem: A Genetic Introduction to Algebraic Num- ber Theory, Springer, 1977. Ellison, W. and F., Th´eorie des nombres, in Abr´eg´e d’Histoire des Math´ematiques 1700–1900, Vol I, (J. Dieudonn´e, ed.) Hermann, Paris, 1978, pp 165–334. Herbrand, J., Le D´eveloppment Moderne de la Th´eorie des Corps Alg´ebriques Corps deClassesetLoisdeR´eciprocit´e, Gauthier-Villars, Paris, 1936. Smith, H.J.S., Report on the the Theory of Numbers, Reports of the British Associ- ation, 1859/1865. (Reprinted by Chelsea, New York, 1965.) Weil, Number Theory: An Approach Through History, Birkh¨auser, 1984. Also: Hasse’s article in Cassels and Fr¨ohlich 1967. Appendix2 in Iyanaga 1975. Iwasawa’s appendixto the Collected Papers of Teiji Takagi. Tate’s article “Problem 9: The general reciprocity law” in Mathematical Developments Arising from Hilbert’s Problems, AMS, 1976. Weil’s article, Oeuvres 1974c. N. Schappacher, On the history of Hilbert’s twelfth problem (preprint). P. Stevenhagan and H.W. Lenstra, Chebo- tar¨ev and his density theorem, Math. Intelligencer, 18.2, 1996, 26–37. For an introduction to nonabelian class field theory: Arthur’s article in 1980 Sem- inar on Harmonic Analysis, CMS Conference Proceedings Vol 1, AMS, 1981. Notations: For a ring R (always with 1), R× denotes the group of invertible elements. We use X =df Y to mean “X is defined to be Y ”or“X = Y by definition”, ∼ X ≈ Y to mean that X and Y are isomorphic, and X = Y to mean X and Y are isomorphic with a given (canonical, or unique) isomorphism. iv Contents Introduction 1 Statement of the main problem, 1; Classification of unramified abelian extensions, 2; Classification of ramified abelian extensions, 2; The Artin map, 4; Explicit class field theory, 6; Nonabelian class field theory, 6; Exercises, 7. Chapter I. Local Class Field Theory 9 1. Statements of the Main Theorems 9 Consequences of Theorems 1.1 and 1.2, 11; Outline of the proof of the main theo- rems, 13. 2. Lubin-Tate Formal Group Laws 15 Power series, 15; Formal group laws, 16; Lubin-Tate group laws, 19. 3. Construction of the extension Kπ of K.22 The local Artin map, 26. 4. The Local Kronecker-Weber Theorem 30 The ramification groups of Kπ,n/K, 30; Upper numbering on ramification groups, 31; The local Kronecker-Weber theorem, 33; The global Kronecker-Weber theorem, 35; Where did it all come from?, 36; Notes, 37. 5. Appendix: Infinite Galois Theory and Inverse Limits 37 Galois theory for infinite extensions, 38; Inverse limits, 40; Chapter II. The Cohomology of Groups 41 1. Cohomology 41 The category of G-modules, 41; Induced modules, 42; Injective G-modules, 43; Definition of the cohomology groups, 44; Shapiro’s lemma, 45; Description of the cohomology groups by means of cochains, 47; The cohomology of L and L×, 49; The cohomology of products, 52; Functorial properties of the cohomology groups, 52; The inflation-restriction exact sequence, 55; Cup-products, 56; 2. Homology; the Tate Groups 57 Definition of the homology groups, 57; The group H1(G, Z), 59; The Tate groups, v vi CONTENTS 60; Cup-products, 61; The cohomology of finite cyclic groups, 61; Tate’s Theorem, 64; 3. The Cohomology of Profinite Groups 66 Direct limits, 66; Profinite groups, 67; Notes, 69; 4. Appendix: Some Homological Algebra 69 Some exact sequences, 69; The language of category theory, 70; Injective objects, 71; Right derived functors, 72; Variants, 75; The Ext groups, 75; References, 76; Chapter III. Local Class Field Theory Continued 77 1. Introduction 77 2. The Cohomology of Unramified Extensions 80 The cohomology of the units, 80; The invariant map, 81; Computation of the local Artin map, 83; 3. The Cohomology of Ramified Extensions 85 4. Complements 87 Alternative description of the local Artin map, 87; The Hilbert symbol, 88; Other Topics, 89; Notes, 89; Chapter IV. Brauer Groups 91 1. Simple Algebras; Semisimple Modules 91 Semisimple modules, 91; Simple k-algebras, 93; Modules over simple k-algebras, 95; 2. Definition of the Brauer Group 96 Tensor products of algebras, 96; Centralizers in tensor products, 97; Primordial elements, 97; Simplicity of tensor products, 98; The Noether-Skolem Theorem, 99; Definition of the Brauer group, 100; Extension of the base field, 101; 3. The Brauer Group and Cohomology 101 Maximal subfields, 102; Central simple algebras and 2-cocycles, 104; 4. The Brauer Groups of Special Fields 107 Finite fields, 107; The real numbers, 108; A nonarchimedean local field, 108; 5. Complements 110 Semisimple algebras, 110; Algebras, cohomology, and group extensions, 110; Brauer groups and K-theory, 111; Notes, 112; Chapter V. Global Class Field Theory: Statements 113 1. Ray Class Groups 113 Ideals prime to S, 113; Moduli, 114; The ray class group, 115; The Frobenius element, 117; CONTENTS vii 2. Dirichlet L-Series and the Density of Primes in Arithmetic Pro- gressions 119 3. The Main Theorems in Terms of Ideals 121 The Artin map, 121; The main theorems of global class field theory, 123; The norm limitation theorem, 125; The principal ideal theorem, 126; The Chebotarev Density Theorem, 128; The conductor-discriminant formula, 129; The reciprocity law and power reciprocity, 130; An elementary unsolved problem, 130; Explicit global class field theory: Kronecker’s Jugentraum and Hilbert’s twelfth problem, 131; Notes, 132; 4. Id`eles 132 Topological groups, 133; Id`eles, 133; Realizing ray class groups as quotients of I, 135; Characters of ideals and of id`eles, 137; Norms of id`eles, 139; 5. The Main Theorms in Terms of Id`eles 140 Example, 143; 6. Appendix: Review of some Algebraic Number Theory 144 The weak approximation theorem, 144; The decomposition of primes, 145; Chapter VI. L-Series and the Density of Primes 147 1. Dirichlet series and Euler products 147 2. Convergence Results 149 Dirichlet series, 149; Euler products, 152; Partial zeta functions; the residue for- mula, 153; 3. Density of the Prime Ideals Splitting in an Extension 155 4. Density of the Prime Ideals in an Arithmetic Progression 157 The Second Inequality, 162; Chapter VII. Global Class Field Theory: Proofs 163 1. Outline 163 2. The Cohomology of the Id`eles 164 The norm map on id´eles, 168; 3. The Cohomology of the Units 169 4. Cohomology of the Id`ele Classes I: the First Inequality 171 5. Cohomology of the Id`ele Classes II: The Second Inequality 174 6. The Algebraic Proof of the Second Inequality 175 7. Application to the Brauer Group 180 8. Completion of the Proof of the Reciprocity Law 182 9. The Existence Theorem 184 viii CONTENTS 10.