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Chapter II Complete Fields
Chapter III Decomposition Groups and the Artin Map
Chapter IV Analytic Methods
Chapter V Class Field Theory
Chapter VI Application of the General Theory to Quadratic Fields
abelian extension algebraic integers algebraic number field archimedean valuation Artin map basis Chapter class field class group class number coefficients completes the proof compute conductor congruence subgroup contains converges Corollary cosets cyclic group Dedekind ring defined denote distinct primes divisible element equal equation equivalent ExERCISE exists factorization finite number finite-dimensional follows fractional ideal Frobenius automorphism function Galois group homomorphism i(Km ideal group imbedding implies infinite prime integral closure isomorphism kernel lattice Lemma Let QI matrix maximal ideal minimum polynomial modulus monic nonzero prime ideal norm number of primes obtain positive integer prime divisor prime ideal primes dividing principal ideal Property Proposition prove Q(Vd quotient field ramification number ramified primes real number reciprocity law holds relative degree relatively prime result root of unity Section sequence set of primes splits completely subfield Suppose Theorem unique unramified valuation ring