Multiplicities of dihedral discriminants

Abstract

Abstract.Given the discriminant dk of a quadratic field k, the number of cyclic relative extensions N\k of fixed odd prime degree p with dihedral absolute Galois group of order 2p , which share a common conductor /, is called the multiplicity of the dihedral discriminant d^ = f2^p~^d^ . In this paper, general formulas for multiplicities of dihedral discriminants are derived by analyzing the p-rank of the ring class group mod / of k . For the special case p = 3 , c4 = -3 , an elementary proof is given additionally. The theory is illustrated by a discussion of all known discriminants of multiplicity > 5 of totally real and complex cubic fields.

THE SECOND p-CLASS GROUP OF A NUMBER FIELD

“…In contrast to [25], where we have solved the multiplicity problem for discriminants of dihedral fields which are ramified with conductor f > 1 over their quadratic subfield, we are now concerned exclusively with unramified extensions having f = 1. For a real quadratic base field K, the cohomology H 0 (Gal(N i |K), U (N i )) of the unit groups U (N i ) with respect to the cyclic Galois groups Gal(N i |K) must be considered to distinguish between partial and total principalisation of K in the N i .…”

Differential principal factors and Pólya property of pure metacyclic fields

“…DPF types of pure quintic fields. We define thirteen possible differential principal factorization types of pure quintic number fields L = Q( 5 √ D), according to the generators of the group Norm N |K U N as the primary invariant and the triplet (A, I, R) defined by the order 5 A := #P a of the group of absolute DPF of L/Q, the index 5 I := (P i : P a ) in the group of intermediate DPF of M/K + , and the index 5 R := (P r : P i ) in the group of relative DPF of N/K, as the secondary invariant (similar to but not identical with our definitions in [19] and in § 4). The connection between the various quantities is given by the chain of equations 5 A+I+R = #P r = (P N/K : P K ) = (E N/K : U σ−1 N ) = 5 · (U K : N N/K (U N )).…”

“…As opposed to the claim P N ∩ I K > P K of Hilbert's Theorem 94 for an unramified cyclic extension N/K of prime degree with conductor f = 1, the subgroup 1 = P N ∩ I K /P K < P N/K /P K , the so-called capitulation kernel of N/K, is trivial for our ramified relative extension with f > 1, because the cyclotomic field K has class number h K = 1. However, the elementary abelian 5-group P r = P N/K /P K , whose generators are principal ideals dividing the relative different of N/K, so-called differential principal factors, consists of three nested subgroups, P r ≥ P i ≥ P a (similar to but not identical with our definitions in [19] and in § 4, since we only want to indicate another slightly different point of view),…”

On quadratic fields with large 3-rank

“…We can count the number of cubic fields of given discriminant in terms of 3-ranks of ring class groups as in Mayer [23]. Unfortunately, it is difficult to use the resulting formulas to decide how likely nonfundamental discriminants are to have a multiplicity of the form (3 r − 1)/2.…”