A Converse of Artin′s Density Theorem: The Case of Cubic Fields Original Research Article

The following problem may be considered as an inverse of Artin′s density theorem: Given n ≥ 2 and p prime, does there exist a density for the set of algebraic integers α of degree n for which p has an assigned splitting in Q(α)? We find that such a set has a density, and we recover the density predicted by Artin′s theorem when p → ∞. Further we given explicit formulae for all splittings in cubic fields.