CLASS FIELD THEORY - James Milne

1 CLASS FIELD the notes for Math 776, University of Michigan, Winter 1997, slightlyrevised from those handed out during the course. They have been substantiallyrevised and expanded from an earlier version, based on my notes from 1993 ( ).My approach to CLASS FIELD THEORY in these notes is eclectic. Although it is possibleto 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 cohomologygroups associated with number fields.

2 When it sheds additional light, I have nothesitated to include more than one proof of a heart of the course is the odd numbered chapters. Chapter II, which is on thecohomology of groups, is basic for the rest of the course, but Chapters IV, VI, andVIII are not essential for reading Chapters III, V, and VII. Except for its first section,Chapter I can be skipped by those not interested inexplicitlocal CLASS FIELD of the form Math xxx are to course notes available send comments and corrections to me at including CLASS FIELD theoryArtin, E.

3 , Algebraic Numbers and Algebraic Functions, NYU,1951.(ReprintedbyGordon 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, , and Fr ohlich, A., (Eds), Algebraic Number THEORY , Academic Press, , , and Vostokov, , Local Fields and their Extensions, AMS, , , Analytic Number THEORY , Prentice Hall, , K., Local CLASS FIELD THEORY , Oxford, , S., (ed.), The THEORY of Numbers, North-Holland, , , Algebraic Number Fields, Academic Press, 1973; Second Edition, AMS, , H.

4 , Number THEORY II, Algebraic Number THEORY , Springer,1992.(Ency-clopaedia of Mathematical Sciences, Vol. 62, Parshin, , and Shafarevich, ,Eds).Lang, S., Algebraic Number THEORY , Addison-Wesley, , J., CLASS FIELD THEORY , Springer, , J., Algebraische Zahlentheorie, Springer, , J-P., Corps Locaux, Hermann, 6, 1997; 1996, 1997, Milne . You may make one copy of these notes for your own , A., Basic Number THEORY , Springer, articles of Serre and Tate in Cassels and Fr ohlich 1967, and Janusz 1973, havebeen especially useful in the writing of these including an introduction to CLASS FIELD theoryCohn, H.

5 , A Classical Invitation to Algebraic Numbers and CLASS Fields, Springer, , , Primes of the Formx2+ny2: Fermat, CLASS FIELD THEORY , and ComplexMultiplication, Wiley, , D. A., Number Fields, for the history of algebraic number THEORY and CLASS FIELD theoryEdwards, , Fermat s Last Theorem: A Genetic Introduction to Algebraic Num-ber THEORY , Springer, , W. and F., Th eorie des nombres, in Abr eg e d Histoire des Math ematiques1700 1900, Vol I, (J. Dieudonn e, ed.) Hermann, Paris,1978, pp 165 , J., Le D eveloppment Moderne de la Th eorie des Corps Alg ebriques CorpsdeClassesetLoisdeR eciprocit e, Gauthier-Villars, Paris, , , Report on the the THEORY of Numbers, Reports of the British Associ-ation,1859/1865.

6 (Reprinted by Chelsea, New York, 1965.)Weil, Number THEORY : An Approach Through History, Birkh auser, : Hasse s article in Cassels and Fr ohlich 1967. Appendi x2 in Iyanaga s appendi xto the Collected Papers of Teiji Takagi. Tate s article Problem9: The general reciprocity law inMathematical Developments Arising from Hilbert sProblems, AMS, 1976. Weil s article, Oeuvres1974c. N. Schappacher, On the historyof Hilbert s twelfth problem (preprint). P. Stevenhagan and Lenstra, Chebo-tar ev and his density theorem, Math.

7 Intelligencer, ,1996, 26 an introduction to nonabelian CLASS FIELD THEORY : Arthur s article in1980 Sem-inar on Harmonic Analysis, CMS Conference Proceedings Vol 1, AMS, :For a ringR(always with 1),R denotes the group of invertibleelements. We useXdf=Yto mean Xis defined to beY or X=Yby definition ,X Yto mean thatXandYare isomorphic, andX =Yto meanXandYareisomorphic with a given (canonical, or unique) of the main problem, 1; Classification of unramified abelian extensions,2; Classification of ramified abelian extensions, 2; The Artin map, 4; Explicit classfield THEORY , 6; Nonabelian CLASS FIELD THEORY , 6; Exercises, I.

8 Local CLASS FIELD Theory91. Statements of the Main Theorems9 Consequences of Theorems and , 11; Outline of the proof of the main theo-rems, Lubin-Tate Formal Group Laws15 Power series, 15; Formal group laws, 16; Lubin-Tate group laws, Construction of the extensionK local Artin map, The Local Kronecker-Weber Theorem30 The ramification groups ofK ,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, Appendix: Infinite Galois THEORY and Inverse Limits37 Galois THEORY for infinite extensions, 38; Inverse limits, 40;Chapter II.

9 The Cohomology of Groups411. Cohomology41 The category ofG-modules, 41; Induced modules, 42; InjectiveG-modules, 43;Definition of the cohomology groups, 44; Shapiro s lemma, 45; Description of thecohomology groups by means of cochains, 47; The cohomology ofLandL , 49; Thecohomology of products, 52; Functorial properties of the cohomology groups, 52; Theinflation-restriction exact sequence, 55; Cup-products, 56;2. Homology; the Tate Groups57 Definition of the homology groups, 57; The groupH1(G,Z), 59; The Tate groups,vviCONTENTS60; Cup-products, 61; The cohomology of finite cyclic groups, 61; Tate s Theorem,64;3.

10 The Cohomology of Profinite Groups66 Direct limits, 66; Profinite groups, 67; Notes, 69;4. Appendix: Some Homological Algebra69 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 Continued771. Introduction772. The Cohomology of Unramified Extensions80 The cohomology of the units, 80; The invariant map, 81; Computation of the localArtin map, 83;3. The Cohomology of Ramified Extensions854.