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Class Field Theory - James Milne

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).

mula for the class number and a density theorem for the primes in an arithmetic progression. He proved the following “unit theorem”: let be a root of a monic irreducible polynomial f.X/with integer coefficients; suppose that f.X/has rreal roots and 2scomplex roots; then Z„ “ is a finitely generated group of rank rCs 1. KUMMER (1810 ...

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