Non-abelian number fields with very large class numbers
نویسندگان
چکیده
منابع مشابه
Class Numbers of Imaginary Abelian Number Fields
Let N be an imaginary abelian number field. We know that hN , the relative class number of N , goes to infinity as fN , the conductor of N , approaches infinity, so that there are only finitely many imaginary abelian number fields with given relative class number. First of all, we have found all imaginary abelian number fields with relative class number one: there are exactly 302 such fields. I...
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Let {Km} be a parametrized family of real abelian number fields of known regulators, e.g. the simplest cubic fields associated with the Q-irreducible cubic polynomials Pm(x) = x −mx2 − (m+ 3)x− 1. We develop two methods for computing the class numbers of these Km’s. As a byproduct of our computation, we found 32 cyclotomic fields Q(ζp) of prime conductors p < 10 for which some prime q ≥ p divid...
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Mathematics Subject Classi cation: Primary, 11R20, 11R29, 11Y40; Secondary, 11M20, 11R42.
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Let {Km} be a parametrized family of simplest real cyclic cubic, quartic, quintic or sextic number fields of known regulators, e.g., the so-called simplest cubic and quartic fields associated with the polynomials Pm(x) = x3 − mx2 − (m + 3)x + 1 and Pm(x) = x4 − mx3 − 6x2 + mx + 1. We give explicit formulas for powers of the Gaussian sums attached to the characters associated with these simplest...
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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.
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ژورنال
عنوان ژورنال: Acta Arithmetica
سال: 2006
ISSN: 0065-1036,1730-6264
DOI: 10.4064/aa125-3-2