Classifying multiplets of totally real cubic fields

TL;DR

This paper extends tables of totally real cubic fields by computing new arithmetical invariants, classifying multiplets based on discriminant and class rank, and verifying a conjecture related to unit indices in ramified extensions.

Contribution

It introduces differential principal factorization types as new invariants and provides a detailed classification scheme for totally real cubic fields based on discriminant, class rank, and conductor.

Findings

Extended existing tables of totally real cubic fields with new invariants.

Classified multiplets according to 3-class rank and prime divisors of conductors.

Refined and verified the Scholz conjecture for ramified extensions.

Abstract

The number of non-isomorphic cubic fields L sharing a common discriminant d(L) = d is called the multiplicity m = m(d) of d. For an assigned value of d, these fields are collected in a multiplet M(d) = (L(1) ,..., L(m)). In this paper, the information in all existing tables of totally real cubic number fields L with positive discriminants d(L) < 10000000 is extended by computing the differential principal factorization types tau(L) in (alpha1, alpha2, alpha3, beta1, beta2, gamma, delta1, delta2, epsilon) of the members L of each multiplet M(d) of non-cyclic fields, a new kind of arithmetical invariants which provide succinct information about ambiguous principal ideals and capitulation in the normal closures N of non-Galois cubic fields L. The classification is arranged with respect to increasing 3-class rank of the quadratic subfields K of the S3-fields N, and to ascending number of prime divisors of the conductor f of N/K. The Scholz conjecture concerning the distinguished index of subfield units (U(N) : U(0)) = 1 for ramified extensions N/K with conductor f > 1 is refined and verified.