A note on $1$-class groups of number fields
نویسندگان
چکیده
منابع مشابه
Number fields with large class groups
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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We establish upper bounds for the smallest height of a generator of a number field k over the rational field Q. Our first bound applies to all number fields k having at least one real embedding. We also give a second conditional result for all number fields k such that the Dedekind zeta-function associated to the Galois closure of k/Q satisfies GRH. This provides a partial answer to a question ...
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The commuting graph of a group is a graph with vertexes set of a subset of a group and two element are adjacent if they commute. The aim of this paper is to obtain the automorphism group of the commuting graph of a conjugacy class in the symmetric groups. The clique number, coloring number, independent number, and diameter of these graphs are also computed.
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Since class numbers of CM number fields of a given degree go to infinity with the absolute values of their discriminants, it is reasonable to ask whether the same conclusion still holds true for the exponents of their ideal class groups. We prove that under the assumption of the Generalized Riemann Hypothesis this is indeed the case. 1991 Mathematics Subject Classification. Primary 11R29, 11R21.
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Following Hasse’s example, various authors have been deriving divisibility properties of minus class numbers of cyclotomic fields by carefully examining the analytic class number formula. In this paper we will show how to generalize these results to CM-fields by using class field theory. Although we will only need some special cases, we have also decided to include a few results on Hasse’s unit...
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ژورنال
عنوان ژورنال: Mathematics of Computation
سال: 1975
ISSN: 0025-5718
DOI: 10.1090/s0025-5718-1975-0409406-1