Cubic field

In mathematics, specifically the area of algebraic number theory, a cubic field is an algebraic number field of degree three.

Definition

If K is a field extension of the rational numbers Q of degree [K:Q] = 3, then K is called a cubic field. Any such field is isomorphic to a field of the form

where f is an irreducible cubic polynomial with coefficients in Q. If f has three real roots, 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 polynomial f. 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.[1]

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 field Q. 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 (in absolute value), namely −108.[2]
  • The complex cubic field obtained by adjoining to Q a root of x3 + x2 − 1 is not pure. It has the smallest discriminant (in absolute value) of all cubic fields, namely −23.[3]
  • Adjoining a root of x3 + x2 − 2x − 1 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.[4]
  • The field obtained by adjoining to Q a root of x3 + x2 − 3x − 1 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.[5]
  • No cyclotomic fields are cubic because the degree of a cyclotomic field is equal to φ(n), where φ is Euler's totient function, which only takes on even values (except for φ(1) = φ(2) = 1).

Galois closure

A cyclic cubic field K is its own Galois closure with Galois group Gal(K/Q) 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(N/Q) is isomorphic to the symmetric group S3 on three letters.

Associated quadratic field

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 (in the case d = 1, the subfield Q is sometimes considered as the "degenerate" quadratic field of discriminant 1). The conductor of N over k is f, and f2 is the relative discriminant of N over K. The discriminant of N is d3f4.[6][7]

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.[7]

Discriminant
The blue crosses are the number of totally real cubic fields of bounded discriminant. The black line is the asymptotic distribution to first order whereas the green line includes the second order term.[8]
The blue crosses are the number of complex cubic fields of bounded discriminant. The black line is the asymptotic distribution to first order whereas the green line includes the second order term.[8]

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 DK satisfies |DK| ≤ N.[9] Formulae are known which calculate the prime decomposition of DK, and so it can be explicitly calculated.[10]

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[11] 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.[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 , according to the totally real or complex case, ζ(s) is the Riemann zeta function, and Γ(s) is the Gamma function. 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.

Unit group

According to Dirichlet's unit theorem, the torsionfree unit rank r of an algebraic number field K 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,[16] which have been interpreted geometrically by Delone and Faddeev.[17]

Notes
  1. Harvey Cohn computed an asymptotic for the number of cyclic cubic fields (Cohn 1954), while Harold Davenport and Hans Heilbronn computed the asymptotic for all cubic fields (Davenport & Heilbronn 1971).
  2. Cohen 1993, §B.3 contains a table of complex cubic fields
  3. Cohen 1993, §B.3
  4. Cohen 1993, §B.4 contains a table of totally real cubic fields and indicates which are cyclic
  5. Cohen 1993, §B.4
  6. Hasse 1930
  7. Cohen 1993, §6.4.5
  8. The exact counts were computed by Michel Olivier and are available at . The first-order asymptotic is due to Harold Davenport and Hans Heilbronn (Davenport & Heilbronn 1971). The second-order term was conjectured by David P. Roberts (Roberts 2001) and a proof has been published by Manjul Bhargava, Arul Shankar, and Jacob Tsimerman (Bhargava, Shankar & Tsimerman 2013).
  9. H. Minkowski, Diophantische Approximationen, chapter 4, §5.
  10. Llorente, P.; Nart, E. (1983). "Effective determination of the decomposition of the rational primes in a cubic field". Proceedings of the American Mathematical Society. 87 (4): 579–585. doi:10.1090/S0002-9939-1983-0687621-6.
  11. Mayer, D. C. (1992). "Multiplicities of dihedral discriminants". Math. Comp. 58 (198): 831–847 and S55–S58. Bibcode:1992MaCom..58..831M. doi:10.1090/S0025-5718-1992-1122071-3.
  12. G. F. Voronoi, Concerning algebraic integers derivable from a root of an equation of the third degree, Master's Thesis, St. Petersburg, 1894 (Russian).
  13. Davenport & Heilbronn 1971
  14. Their work can also be interpreted as a computation of the average size of the 3-torsion part of the class group of a quadratic field, and thus constitutes one of the few proven cases of the Cohen–Lenstra conjectures: see, e.g. Bhargava, Manjul; Varma, Ila (2014), The mean number of 3-torsion elements in the class groups and ideal groups of quadratic orders, arXiv:1401.5875, Bibcode:2014arXiv1401.5875B, This theorem [of Davenport and Heilbronn] yields the only two proven cases of the Cohen-Lenstra heuristics for class groups of quadratic fields.
  15. Roberts 2001, Conjecture 3.1
  16. Voronoi, G. F. (1896). On a generalization of the algorithm of continued fractions (in Russian). Warsaw: Doctoral Dissertation.
  17. Delone, B. N.; Faddeev, D. K. (1964). The theory of irrationalities of the third degree. Translations of Mathematical Monographs. 10. Providence, Rhode Island: American Mathematical Society.
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References
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