The number of non-isomorphic cubic fields L sharing a common discriminant dL = d is called the multiplicity m = m(d) of d. For an assigned value of d, these fields are collected in a homogeneous multiplet M d = (L1, . . . , Lm). By entirely new techniques for the construction and classification, we determine the differential principal factorization types τ (Li) ∈ {α1, α2, α3, β1, β2, γ, δ1, δ2, ε} of the members Li of each multiplet M d of non-cyclic totally real cubic fields with discriminants

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... d < 10 7 . This is 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 (UN : U0) = 1 for ramified extensions N/K with conductor f > 1 is also verified by a tremendous minimal discriminant d = 18 251 060 outside of the systematic range d < 10 7 .