On the Computation of Invariants in non-Normal, non-Pure Cubic Fields and in Their Normal Closures

Let K=Q(theta) be the algebraic number field formed by adjoining theta to the rationals where theta is a real root of an irreducible monic cubic polynomial f(x) in Z[x]. If theta is not the cube root of a rational integer, we call the field K a non-pure cubic field, and if K doesn't contain the conjugates of theta, we call K a non-normal cubic field. A method described by Martinet and Payan allows us to construct such fields from elements of a quadratic field. In this work, we examine such non-normal, non-pure cubic fields and their normal closures, using algorithms in Mathematica to compute various invariants of these fields. In addition, we prove general results relating the ranks of the ideal class groups of the rings of integers of these cubic fields to those of their normal closures.