Cubic Field - Discriminant

Discriminant

Since the sign of the discriminant of a number field K is (−1)r₂, where r₂ is the number of conjugate pairs of complex embeddings of K into C, the discriminant of a cubic field will be positive precisely when the field is totally real, and negative if it is a complex cubic field.

Given some real number N > 0 there are only finitely many cubic fields K whose discriminant D_K satisfies |D_K| ≤ N. Formulae are known which calculate the prime decomposition of D_K, and so it can be explicitly calculated.

However, it should be pointed out that, different from quadratic fields, several non-isomorphic cubic fields K₁, ..., K_m may share the same discriminant D. The number m of these fields is called the multiplicity of the discriminant D. Some small examples are m = 2 for D = −1836,3969, m = 3 for D = −1228,22356, m = 4 for D = −3299,32009, and m = 6 for D = −70956,3054132.

Any cubic field K will be of the form K = Q(θ) for some number θ that is a root of the irreducible polynomial

with a and b both being integers. The discriminant of f is Δ = 4a3 − 27b2. Denoting the discriminant of K by D, the index i(θ) of θ is then defined by Δ = i(θ)2D.

In the case of a non-cyclic cubic field K this index formula can be combined with the conductor formula D = f2d to obtain a decomposition of the polynomial discriminant Δ = i(θ)2f2d into the square of the product i(θ)f and the discriminant d of the quadratic field k associated with the cubic field K, where d is squarefree up to a possible factor 22 or 23. Georgy Voronoy gave a method for separating i(θ) and f in the square part of Δ.

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). 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:

where A_± = 1 or 3, B_± = 1 or, according to the totally real or complex case, ζ(s) is the Riemann zeta function, and Γ(s) is the Gamma function. A proof of this formula has been announced by Bhargava, Shankar & Tsimerman (2010) using methods based on Bhargava's earlier work, as well as Taniguchi & Thorne (2011) based on the Shintani zeta function.