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arXiv · 2102.12187

Classifying multiplets of totally real cubic fields

Daniel C. Mayer
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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.

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Daniel C. Mayer. 2021-02-24. Classifying multiplets of totally real cubic fields. https://arxiv.org/abs/2102.12187

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