This chapter is devoted to the arithmetic of Abelian extensions of the rationals, i.e., normal extensions K/ℚ with an Abelian Galois group. According to the Kronecker-Weber theorem (Theorem 6.18) every such extension is contained in a suitable cyclotomic field K n = ℚ(ζn). The least integer f with the property K⊂K f is called the conductor of K, and is denoted by f(K).S The main properties of the conductor are listed in the following proposition:
KeywordsPrime Ideal Galois Group Bernoulli Number Quadratic Field Abelian Extension
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