Distribution of Units of a Cubic Field with Negative Discriminant Original Research Article

Let F be a cubic number field with negative discriminant. Taking into account the extension degree of the ray class field modulo a prime ideal, we study the residual index I(image) of residue classes represented by units in the multiplicative group of the residue field modulo a prime ideal image. The possible minimal value ℓ(image) is given, and we give the density of prime ideals image with I(image)=ℓ(image) if the degree of image is one, and a conjecture if the degree is 2 or 3.