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After a review of the quadratic case, a general problem about the existence of number fields of a fixed degree with extremely large class numbers is formulated. This problem is solved for abelian cubic fields. Then some conditional results proven elsewhere are discussed about totally real number fields of a fixed degree, each of whose normal closure has the symmetric group as Galois group. 1 Introduction. It was Littlewood who first addressed the question of how large the class number h of an imaginary quadratic field Q ( √ d) can be as a function of |d | as d → − ∞ through fundamental discriminants. In 1927 [14] he showed, assuming the generalized Riemann hypothesis (GRH), that for all fundamental d <

Year: 2009

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