5-Class towers of cyclic quartic fields arising from quintic reflection

TL;DR

This paper investigates the structure and length of 5-class towers of certain cyclic quartic fields derived from quintic reflection, providing theoretical insights supported by extensive computational data.

Contribution

It characterizes the Galois groups and class tower lengths of specific cyclic quartic fields associated with quintic reflection, extending understanding of their unramified extensions.

Findings

Determined the possible Galois group structures for these fields.

Established the exact length of the 5-class towers in many cases.

Performed extensive computations for a large range of discriminants.

Abstract

Let zeta5 be a primitive fifth root of unity and d<>1 be a quadratic fundamental discriminant not divisible by 5. For the 5-dual cyclic quartic field M=Q((zeta5-zeta5^-1)*d^1/2) of the quadratic fields k1=Q(d^1/2) and k2=Q((5*d)^1/2) in the sense of the quintic reflection theorem, the possibilities for the isomophism type of the Galois group G(5,2)M=Gal(M(5,2)/M) of the second Hilbert 5-class field M(5,2) of M are investigated, when the 5-class group Cl5(M) is elementary bicyclic of rank two. Usually, the maximal unramified pro-5-extension M(5,infinity) of M coincides with M(5,2) already. The precise length ell5(M) of the 5-class tower of M is determined, when G(5,2)M is of order less than or equal to 5^5. Theoretical results are underpinned by the actual computation of all 83, respectively 93, cases in the range 0<d<10^4, respectively -2*10^5<d<0.