Numerical Results on Class Groups of Imaginary Quadratic Fields.
01/2006; pp.87-101 In proceeding of: Algorithmic Number Theory, 7th International Symposium, ANTS-VII, Berlin, Germany, July 23-28, 2006, Proceedings
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ABSTRACT: Let p denote an odd prime. For all p-admissible conductors c over a quadratic number field K = Q(√ d), p-ring spaces Vp(c) modulo c are introduced by defining a morphism ψ : f → Vp(f) from the divisor lattice N of positive integers to the lattice S of subspaces of the direct product Vp of the p-elementary class group C/C p and unit group U/U p of K. Their properties admit an exact count of all extension fields N over K, having the dihedral group of order 2p as absolute Galois group Gal(N |Q) and sharing a common discriminant d N and conductor c over K. The number mp(d, c) of these extensions is given by a formula in terms of positions of p-ring spaces in S, whose complexity increases with the dimension of the vector space Vp over the finite field Fp, called the modified p-class rank σp of K. Up to now, explicit multiplicity formulas for discriminants were known for quadratic fields with 0 ≤ σp ≤ 1 only. Here, the results are extended to σp = 2, underpinned by concrete numerical examples.Journal of Number Theory 09/2012; · 0.56 Impact Factor
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