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abelian extension adele algebraic integers archimedean absolute values Artin map assertion assume automorphism bounded Chapter character class field theory coefficients compact complex numbers concludes the proof conjugate contained converges Corollary cyclic cyclotomic decomposition group Dedekind ring define degree denote Dirichlet series discriminant embedded equal factor group finite extension finite number follows formula Fourier fractional ideal functional equation fundamental domain Galois extension Galois group given group G Hence homomorphism ideal class group idele induced inequality integral closure isomorphism kernel L-series lattice Lemma Let f Let K/k maximal ideal module multiplicative group non-zero number field number of elements obtain open subgroup p-adic positive integer prime ideal prime number Proposition prove quasi-character quotient field Re(s residue class field roots of unity satisfies splits completely subset surjective Theorem trivial unit unramified variables vector whence write zeta function