Cubic field

It is a rare occurrence in the sense that if the set of cubic fields is ordered by discriminant, then the proportion of cubic fields which are cyclic approaches zero as the bound on the discriminant approaches infinity.

Georgy Voronoy gave a method for separating i(θ) and f in the square part of Δ.

[12] The study of the number of cubic fields whose discriminant is less than a given bound is a current area of research.

Let N+(X) (respectively N−(X)) denote the number of totally real (respectively complex) cubic fields whose discriminant is bounded by X in absolute value.

In the early 1970s, Harold Davenport and Hans Heilbronn determined the first term of the asymptotic behaviour of N±(X) (i.e. as X goes to infinity).

[13][14] By means of an analysis of the residue of the Shintani zeta function, combined with a study of the tables of cubic fields compiled by Karim Belabas (Belabas 1997) and some heuristics, David P. Roberts conjectured a more precise asymptotic formula:[15] where A± = 1 or 3, B± = 1 or

Proofs of this formula have been published by Bhargava, Shankar & Tsimerman (2013) using methods based on Bhargava's earlier work, as well as by Taniguchi & Thorne (2013) based on the Shintani zeta function.

These fundamental systems of units can be calculated by means of generalized continued fraction algorithms by Voronoi,[16] which have been interpreted geometrically by Delone and Faddeev.