If K is a field extension of the rational numbersQ of degree = 3, then K is called a cubic field. Any such field is isomorphic to a field of the form where f is an irreducible cubicpolynomial with coefficients in Q. If f has three realroots, then K is called a totally real cubic field and it is an example of a totally real field. If, on the other hand, f has a non-real root, then K is called a complex cubic field. A cubic field K is called a cyclic cubic field, if it contains all three roots of its generating polynomialf. Equivalently, K is a cyclic cubic field if it is a Galois extension of Q, in which case its Galois group over Q is cyclic of order three. This can only happen if K is totally real. 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. A cubic field is called a pure cubic field, if it can be obtained by adjoining the real cube root of a cubefree positive integer n to the rational number fieldQ. Such fields are always complex cubic fields since each positive number has two complex non-real cube roots.
Examples
Adjoining the real cube root of 2 to the rational numbers gives the cubic field. This is an example of a pure cubic field, and hence of a complex cubic field. In fact, of all pure cubic fields, it has the smallest discriminant, namely −108.
The complex cubic field obtained by adjoining to Q a root of is not pure. It has the smallest discriminant of all cubic fields, namely −23.
Adjoining a root of to Q yields a cyclic cubic field, and hence a totally real cubic field. It has the smallest discriminant of all totally real cubic fields, namely 49.
The field obtained by adjoining to Q a root of is an example of a totally real cubic field that is not cyclic. Its discriminant is 148, the smallest discriminant of a non-cyclic totally real cubic field.
A cyclic cubic field K is its own Galois closure with Galois group Gal isomorphic to the cyclic group of order three. However, any other cubic field K is a non-galois extension of Q and has a field extension N of degree two as its Galois closure. The Galois group Gal is isomorphic to the symmetric groupS3 on three letters.
The discriminant of a cubic field K can be written uniquely as df2 where d is a fundamental discriminant. Then, K is cyclic if, and only if, d = 1, in which case the only subfield of K is Q itself. If d ≠ 1, then the Galois closure N of K contains a unique quadratic field k whose discriminant is d. The conductor of N over k is f, and f2 is the relative discriminant of N over K. The discriminant of N is d3f4. The field K is a pure cubic field if, and only if, d = −3. This is the case for which the quadratic field contained in the Galois closure of K is the cyclotomic field of cube roots of unity.
Discriminant
Since the sign of the discriminant of a number fieldK is 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 DK satisfies |DK| ≤ N. Formulae are known which calculate the prime decomposition of DK, and so it can be explicitly calculated. Unlike quadratic fields, several non-isomorphic cubic fields K1,..., Km 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 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 indexi of θ is then defined by Δ = i2D. 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 = i2f2d into the square of the product if 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 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+ denote the number of totally real 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±. 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 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 is the Riemann zeta function, and Γ is the Gamma function. Proofs of this formula have been published by using methods based on Bhargava's earlier work, as well as by based on the Shintani zeta function.
Unit group
According to Dirichlet's unit theorem, the torsionfree unit rank r of an algebraic number fieldK with r1 real embeddings and r2 pairs of conjugate complex embeddings is determined by the formula r = r1 + r2 − 1. Hence a totally real cubic field K with r1 = 3, r2 = 0 has two independent units ε1, ε2 and a complex cubic field K with r1 = r2 = 1 has a single fundamental unit ε1. These fundamental systems of units can be calculated by means of generalized continued fraction algorithms by Voronoi, which have been interpreted geometrically by Delone and Faddeev.
