Class Field Theory - James Milne

1 Class Field MilneVersion 23, 2013 Class Field Theory describes the abelian extensions of a local or global Field in terms of thearithmetic of the Field itself. These notes contain an exposition of abelian Class Field theoryusing the algebraic/cohomological approach of Chevalley and Artin and Tate. The explicitapproach of Lubin and Tate in the local case and the analytic approach in the global caseare also explained. The original version of the notes was distributed during the teaching ofan advanced graduate Milne },title={ Class Field Theory ( )},year={2013},note={Available at },pages={281+viii}} (August 14, 1996). First version on the (May 6, 1997). Substantially revised and expanded; 222 (March 2, 2008). Corrected, revised, and expanded; 287 (May 30, 2011). Many minor fixes; 287 (March 23, 2013). Minor fixes and improvements; 289 at send comments and corrections to me at the address on my web photograph is of Mt Christina from the McKellar Saddle, New 1996, 1997, 2008, 2011, 2013, paper copies for noncommercial personal use may be made without explicit permis-sion from the copyright Class Field Theory : Lubin-Tate Theory191 Statements of the Main Theorems.

2 192 Lubin-Tate Formal Group Laws ..273 Construction of the extensionK ofK..354 The Local Kronecker-Weber Theorem ..43 AAppendix: Infinite Galois Theory and Inverse Limits ..50 IIThe Cohomology of Groups551 Cohomology ..552 Homology ..723 The Tate groups ..754 The Cohomology of Profinite Groups ..83 AAppendix: Some Homological Algebra ..86 III Local Class Field Theory : Cohomology951 The Cohomology of Unramified Extensions ..952 The Cohomology of Ramified Extensions .. 1013 The Local Artin Map .. 1054 The Hilbert symbol .. 1085 The Existence Theorem .. 113IV Brauer Groups1171 Simple Algebras; Semisimple Modules .. 1172 Definition of the Brauer Group .. 1233 The Brauer Group and Cohomology .. 1294 The Brauer Groups of Special Fields .. 1365 Complements .. 139 VGlobal Class Field Theory : Statements1431 Ray Class Groups .. 1432L-series .. 1503 The Main Theorems in Terms of Ideals .. 1524Id`eles .. 1645 The Main Theorms in Terms of Id`eles.

3 172 VIL-Series and the Density of Primes179iii1 Dirichlet series and Euler products .. 1792 Convergence Results .. 1813 Density of the Prime Ideals Splitting in an Extension .. 1874 Density of the Prime Ideals in an Arithmetic Progression .. 189 VII Global Class Field Theory : Proofs1971 Outline .. 1972 The Cohomology of the Id`eles .. 1993 The Cohomology of the Units .. 2034 Cohomology of the Id`ele Classes I: the First Inequality .. 2065 Cohomology of the Id`ele Classes II: The Second Inequality .. 2086 The Algebraic Proof of the Second Inequality .. 2107 Application to the Brauer Group .. 2158 Completion of the Proof of the Reciprocity Law .. 2179 The Existence Theorem .. 220 AAppendix: Kummer Theory .. 222 VIII Complements2251 When are localnth powers globalnth powers? .. 2252 The Grunwald-Wang Theorem .. 2273 The local-global principle for norms and quadratic forms .. 2304 The Fundamental Exact Sequence and the Fundamental Class .

4 2355 Higher Reciprocity Laws .. 2396 The Classification of Quadratic Forms over a Number Field .. 2467 Density Theorems .. 2558 Function Fields .. 2579 Cohomology of Number Fields .. 25710 More onL-series .. 257 AExercises261 BSolutions to Exercises265 CSources for the history of Class Field use the standard (Bourbaki) notations:ND f0;1;2;:::g;ZDring of integers,QDfield of rational numbers,RDfield of real numbers,CDfield of complex numbers,FpDZ=pZDfield withpelements,pa prime integersmandn,mjnmeans thatmdividesn, ,n2mZ. Throughout the notes,pis a prime number, ,pD2;3;5;:::.Given an equivalence relation, denotes the equivalence Class containing . Theempty set is denoted by;. The cardinality of a setSis denoted byjSj(sojSjis the numberof elements inSwhenSis finite). LetIandAbe sets; a family of elements ofAindexedbyI, , is a functioni7!aiWI! Y Xis a subset ofY(not necessarily proper);XdefDY Xis defined to beY, or equalsYby definition;X Y Xis isomorphic toY;X'Y XandYare canonically isomorphic (or there is a given or unique isomorphism);,!

5 Denotes an injective map; denotes a surjective is standard to use Gothic (fraktur) letters for ideals:a b c m n p q A B C M N P Qa b c m n p q A B C M N P QPREREQUISITESThe algebra usually covered in first-year graduate courses and a course in algebraic numbertheory, for example, my course notes listed addition to the references listed at the end (and in footnotes), I shall refer to the followingof my course notes:GTGroup Theory ( , 2013)FTFields and Galois Theory ( , 2013)ANTA lgebraic Number Theory ( , 2013).ACKNOWLEDGEMENTSI thank the following for providing corrections and comments for earlier versions of thesenotes: Vincenzo Acciaro; Tom Bachmann; Oliver Braeunling; Chen, Bingxu; KwanghoChoiy; Brian Conrad; Keith Conrad; Giuseppe Canuto; Ross Griffiths; Darij Grinberg;Florian Herzig; Chong Hui; Herv e Jacquet; Timo Keller; Keenan Kidwell; Michiel Kosters;Tyler Lawson; Franz Lemmermeyer; Kim Nguyen; Catherine O Neil; Kartik Prasanna;Nandini Ranganathan; Peter Roquette; Joshua Seaton; Corinne Sheridan; Jonah Sinick;Daniel Sparks; Yu Zhao; and have been reading Chevalley s new book on Class Field Theory ; I am not reallydoing research, just trying to cultivate , FERMAT(1601 1665).

6 Stated his last theorem , and proved it formD4. He also posedthe problem of finding integer solutions to the equation,X2 AY2D1; A2Z;(1)which is essentially the problem1of finding the units inZ pA . The English mathemati-cians found an algorithm for solving the problem, but neglected to prove that the algorithmalways (1707 1783). He introduced analysis into the study of the prime numbers, and hediscovered an early version of the quadratic reciprocity (1736 1813). He found the complete form of the quadratic reciprocity law: pq qp D. 1/.p 1/.q 1/=4; p;qodd primes,and he proved that the algorithm for solving (1) always leads to a solution,LEGENDRE(1752 1833). He introduced the Legendre symbol mp , and gave an incom-plete proof of the quadratic reciprocity law. He proved the following local-global principlefor quadratic forms in three variables overQ: a quadratic ;Y;Z/has a nontrivialzero inQif and only if it has one inRand the congruenceQ 0modpnhas a nontrivialsolution for (1777 1855).

7 He found the first complete proofs of the quadratic reciprocity studied the Gaussian integersZ i in order to find a quartic reciprocity law. He studiedthe classification of binary quadratic forms overZ, which is closely related to the problemof finding the Class numbers of quadratic (1805 1859). He introducedL-series, and used them to prove an analytic for-mula for the Class number and a density theorem for the primes in an arithmetic proved the following unit theorem : let be a root of a monic irreducible integer coefficients; suppose roots and2scomplex roots;thenZ is a finitely generated group of rankrCs (1810 1893). He made a deep study of the arithmetic of cyclotomic fields, mo-tivated by a search for higher reciprocity laws, and showed that unique factorization couldbe recovered by the introduction of ideal numbers . He proved that Fermat s last theoremholds for regular (1822 1901). He made important contributions to quadratic forms, and he showedthat the roots of a polynomial of degree5can be expressed in terms of elliptic (1823 1852).

8 He published the first complete proofs for the cubic and quarticreciprocity (1823 1891). He developed an alternative to Dedekind s ideals. He also hadone of the most beautiful ideas in mathematics for generating abelian extensions of numberfields (the Kronecker liebster Jugendtraum).1 The Indian mathematician Bhaskara (12th century) knew general rules for finding solutions to the discovered [the quadratic reciprocity law] and, apparently, it is possible to construct a proof of thetheorem using different fragments that can be found in Euler s Opera omnia or his Nachless. It was Gauss whogave the first complete proof.. (Michael Berg, MR2131680).viiRIEMANN(1826 1866). Studied the Riemann zeta function, and made the Riemann (1831 1916). He laid the modern foundations of algebraic number Theory byfinding the correct definition of the ring of integers in a number Field , by proving that idealsfactor uniquely into products of prime ideals in such rings, and by showing that, moduloprincipal ideals, they fall into finitely many classes.

9 Defined the zeta function of a (1842 1913). He found the correct generalization of Class group to allow forramification. Made important progress in Class Field Theory and the Kronecker (1861 1941). He defined the Field ofp-adic numbers (as the set of infinite sumsP1nD kanpn,an2f0;1;:::;p 1g) in the (1862 1943). He wrote a very influential book on algebraic number Theory in1897, which gave the first systematic account of the Theory . Some of his famous problemswere on number Theory , and have also been (1875 1960). He proved the fundamental theorems of abelian Class Field Theory ,as conjectured by Weber and (1882 1935). Together with Artin, she laid the foundations of modern algebrain which axioms and conceptual arguments are emphasized, and she contributed to theclassification of central simple algebras over number (1887 1947). Introduced HeckeL-series generalizing both Dirichlet sL-series andDedekind s zeta (1898 1962). He found the Artin reciprocity law , which is the main theorem ofclass Field Theory (improvement of Takagi s results).

10 Introduced the (1898 1979). He gave the first proof of local Class Field Theory , proved the Hasse(local-global) principle for all quadratic forms over number fields, and contributed to theclassification of central simple algebras over number (1901 1977). Defined the Brauer group, and contributed to the classification ofcentral simple algebras over number (1906 1998). Defined the Weil group, which enabled him to give a common gener-alization of ArtinL-series and (1909 84). The main statements of Class Field Theory are purely algebraic,but all the earlier proofs used analysis; Chevalley gave a purely algebraic proof. With hisintroduction of id`eles he was able to give a natural formulation of Class Field Theory forinfinite abelian (1917 1998). He introduced an important new approach into algebraic numbertheory which was suggested by the Theory of curves over finite (1925 ). He proved new results in group cohomology, which allowed him to givean elegant reformulation of Class Field Theory .