  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.
