A rational genus, class number divisibility, and unit theory for pure cubic fields

Abstract The classical genus theory of Gauss has been extended by Hilbert from the quadratic field over the rational field to the Kummer field over the cyclotomic field as a precise analog. Still, many problems involving the computation of pure cubic unit and class number are related to the rational and not the cubic cyclotomic field. It is, therefore, desirable to have (at least partially) a genus theory of cubic characters and an analog of such things as Hilbert's Theorem 90, ambiguous (principal) classes, and class group invariants divisible by 3 and 9,-all based on the rational field. This study shows a strong similarity between pure cubic and quadratic fields.