Class Field Theory
This classic book, originally published in 1968, is based on notes of a year-long seminar the authors ran at Princeton University. The primary goal of the book was to give a rather complete presentation of algebraic aspects of global class field theory ... In this revised edition, two mathematical additions complementing the exposition of the original text are made. The new edition also contains several new footnotes, additional references, and historical comments.
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The First Fundamental Inequality
Second Fundamental Inequality
The Existence Theorem
Connected Component of Idele Classes
The GrunwaldWang Theorem
Higher Ramification Theory
2-cocycle a e G abelian extension algebraic archimedean primes automorphism Axiom Brauer group Chapter character class field theory class formation cocycle class cohomology groups commutative compact concludes the proof consequently contains COROLLARY corresponding cyclic extension defined definition denote extension of degree factor group finite extension finite index function fields fundamental class Galois group GK/F global field group extension group G Hence homomorphism idele class induced integer invariant invp irreducible isomorphism kernel Lemma Let G Let K/k module multiplicative group n-th power n-th roots neighborhood norm residue symbol norm subgroup normal extension normal layer K/F number fields obtain open subgroup ordx prime degree properties PROPOSITION prove reciprocity law residue class field root of unity second inequality set of primes shows subfield subgroup of finite subgroup of G trivial unramified W-diagram Weil group