Cyclic Cubic Field
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In mathematics, specifically the area of algebraic number theory, a cubic field is an
algebraic number field In mathematics, an algebraic number field (or simply number field) is an extension field K of the field of rational numbers such that the field extension K / \mathbb has finite degree (and hence is an algebraic field extension). Thus K is a ...
of
degree Degree may refer to: As a unit of measurement * Degree (angle), a unit of angle measurement ** Degree of geographical latitude ** Degree of geographical longitude * Degree symbol (°), a notation used in science, engineering, and mathemati ...
three.


Definition

If ''K'' is a
field extension In mathematics, particularly in algebra, a field extension is a pair of fields E\subseteq F, such that the operations of ''E'' are those of ''F'' restricted to ''E''. In this case, ''F'' is an extension field of ''E'' and ''E'' is a subfield of ...
of the
rational number In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all ra ...
s Q of
degree Degree may refer to: As a unit of measurement * Degree (angle), a unit of angle measurement ** Degree of geographical latitude ** Degree of geographical longitude * Degree symbol (°), a notation used in science, engineering, and mathemati ...
'K'':Qnbsp;= 3, then ''K'' is called a cubic field. Any such field is isomorphic to a field of the form :\mathbf (f(x)) where ''f'' is an irreducible
cubic polynomial In mathematics, a cubic function is a function of the form f(x)=ax^3+bx^2+cx+d where the coefficients , , , and are complex numbers, and the variable takes real values, and a\neq 0. In other words, it is both a polynomial function of degree ...
with
coefficient In mathematics, a coefficient is a multiplicative factor in some term of a polynomial, a series, or an expression; it is usually a number, but may be any expression (including variables such as , and ). When the coefficients are themselves ...
s in Q. If ''f'' has three
real Real may refer to: Currencies * Brazilian real (R$) * Central American Republic real * Mexican real * Portuguese real * Spanish real * Spanish colonial real Music Albums * ''Real'' (L'Arc-en-Ciel album) (2000) * ''Real'' (Bright album) (201 ...
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 In mathematics, a Galois extension is an algebraic field extension ''E''/''F'' that is normal and separable; or equivalently, ''E''/''F'' is algebraic, and the field fixed by the automorphism group Aut(''E''/''F'') is precisely the base fiel ...
of Q, in which case its
Galois group In mathematics, in the area of abstract algebra known as Galois theory, the Galois group of a certain type of field extension is a specific group associated with the field extension. The study of field extensions and their relationship to the po ...
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 In mathematics, the discriminant of a polynomial is a quantity that depends on the coefficients and allows deducing some properties of the roots without computing them. More precisely, it is a polynomial function of the coefficients of the ori ...
, 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 \sqrt /math> of a cube-free positive integer ''n'' to the rational number field Q. Such fields are always complex cubic fields since each positive number has two
complex Complex commonly refers to: * Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe ** Complex system, a system composed of many components which may interact with each ...
non-real cube roots.


Examples

*Adjoining the real cube root of 2 to the rational numbers gives the cubic field \mathbf(\sqrt . 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. *The complex cubic field obtained by adjoining to Q a root of is not pure. It has the smallest discriminant (in absolute value) 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. *No
cyclotomic field In number theory, a cyclotomic field is a number field obtained by adjoining a complex root of unity to , the field of rational numbers. Cyclotomic fields played a crucial role in the development of modern algebra and number theory because o ...
s are cubic because the degree of a cyclotomic field is equal to φ(''n''), where φ is
Euler's totient function In number theory, Euler's totient function counts the positive integers up to a given integer that are relatively prime to . It is written using the Greek letter phi as \varphi(n) or \phi(n), and may also be called Euler's phi function. In ...
, 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 In group theory, a branch of abstract algebra in pure mathematics, a cyclic group or monogenous group is a group, denoted C''n'', that is generated by a single element. That is, it is a set of invertible elements with a single associative bi ...
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 In abstract algebra, the symmetric group defined over any set is the group whose elements are all the bijections from the set to itself, and whose group operation is the composition of functions. In particular, the finite symmetric group ...
''S''3 on three letters.


Associated quadratic field

The discriminant of a cubic field ''K'' can be written uniquely as ''df''2 where ''d'' is a
fundamental discriminant In mathematics, a fundamental discriminant ''D'' is an integer invariant in the theory of integral binary quadratic forms. If is a quadratic form with integer coefficients, then is the discriminant of ''Q''(''x'', ''y''). Conversely, every integ ...
. Then, ''K'' is cyclic
if and only if In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false. The connective is bi ...
''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 In algebraic number theory, a quadratic field is an algebraic number field of degree two over \mathbf, the rational numbers. Every such quadratic field is some \mathbf(\sqrt) where d is a (uniquely defined) square-free integer different from 0 ...
''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 Conductor or conduction may refer to: Music * Conductor (music), a person who leads a musical ensemble, such as an orchestra. * ''Conductor'' (album), an album by indie rock band The Comas * Conduction, a type of structured free improvisation ...
of ''N'' over ''k'' is ''f'', and ''f''2 is the
relative discriminant In mathematics, the discriminant of an algebraic number field is a numerical invariant that, loosely speaking, measures the size of the (ring of integers of the) algebraic number field. More specifically, it is proportional to the squared volume ...
of ''N'' over ''K''. The discriminant of ''N'' is ''d''3''f''4. 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 In mathematics, a root of unity, occasionally called a de Moivre number, is any complex number that yields 1 when raised to some positive integer power . Roots of unity are used in many branches of mathematics, and are especially important i ...
.


Discriminant

Since the sign of the
discriminant In mathematics, the discriminant of a polynomial is a quantity that depends on the coefficients and allows deducing some properties of the roots without computing them. More precisely, it is a polynomial function of the coefficients of the ori ...
of a number field ''K'' is (−1)''r''2, where ''r''2 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. Unlike quadratic fields, several non-isomorphic cubic fields ''K''1, ..., ''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(θ) for some number θ that is a root of an irreducible polynomial :f(X)=X^3-aX+b where ''a'' and ''b'' are integers. The
discriminant In mathematics, the discriminant of a polynomial is a quantity that depends on the coefficients and allows deducing some properties of the roots without computing them. More precisely, it is a polynomial function of the coefficients of the ori ...
of ''f'' is Δ = 4''a''3 − 27''b''2. Denoting the discriminant of ''K'' by ''D'', the index ''i''(θ) of θ is then defined by Δ = ''i''(θ)2''D''. In the case of a non-cyclic cubic field ''K'' this index formula can be combined with the conductor formula ''D'' = ''f''2''d'' to obtain a decomposition of the polynomial discriminant Δ = ''i''(θ)2''f''2''d'' 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 In mathematics, a square-free integer (or squarefree integer) is an integer which is divisible by no square number other than 1. That is, its prime factorization has exactly one factor for each prime that appears in it. For example, is square-f ...
up to a possible factor 22 or 23.
Georgy Voronoy Georgy Feodosevich Voronoy (russian: Георгий Феодосьевич Вороной; ukr, Георгій Феодосійович Вороний; 28 April 1868 – 20 November 1908) was an Imperial Russian mathematician of Ukrainian descent ...
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 Harold Davenport FRS (30 October 1907 – 9 June 1969) was an English mathematician, known for his extensive work in number theory. Early life Born on 30 October 1907 in Huncoat, Lancashire, Davenport was educated at Accrington Grammar Scho ...
and
Hans Heilbronn Hans Arnold Heilbronn (8 October 1908 – 28 April 1975) was a mathematician. Education He was born into a German-Jewish family. He was a student at the universities of Berlin, Freiburg and Göttingen, where he met Edmund Landau, who supervised ...
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 and some
heuristic A heuristic (; ), or heuristic technique, is any approach to problem solving or self-discovery that employs a practical method that is not guaranteed to be optimal, perfect, or rational, but is nevertheless sufficient for reaching an immediat ...
s, David P. Roberts
conjecture In mathematics, a conjecture is a conclusion or a proposition that is proffered on a tentative basis without proof. Some conjectures, such as the Riemann hypothesis (still a conjecture) or Fermat's Last Theorem (a conjecture until proven in 1 ...
d a more precise asymptotic formula: :N^\pm(X)\sim\fracX+\fracX^ where ''A''± = 1 or 3, ''B''± = 1 or \sqrt, according to the totally real or complex case, ζ(''s'') is the Riemann zeta function, and Γ(''s'') is the
Gamma function In mathematics, the gamma function (represented by , the capital letter gamma from the Greek alphabet) is one commonly used extension of the factorial function to complex numbers. The gamma function is defined for all complex numbers except th ...
. 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 In mathematics, Dirichlet's unit theorem is a basic result in algebraic number theory due to Peter Gustav Lejeune Dirichlet. It determines the rank of the group of units in the ring of algebraic integers of a number field . The regulator i ...
, the torsion-free unit rank ''r'' of an algebraic number field ''K'' with ''r''1 real embeddings and ''r''2 pairs of conjugate complex embeddings is determined by the formula ''r'' = ''r''1 + ''r''2 − 1. Hence a totally real cubic field ''K'' with ''r''1 = 3, ''r''2 = 0 has two independent units ε1, ε2 and a complex cubic field ''K'' with ''r''1 = ''r''2 = 1 has a single fundamental unit ε1. These fundamental systems of units can be calculated by means of generalized
continued fraction In mathematics, a continued fraction is an expression obtained through an iterative process of representing a number as the sum of its integer part and the reciprocal of another number, then writing this other number as the sum of its integ ...
algorithms by Voronoi, which have been interpreted geometrically by Delone and Faddeev.


Notes


References

*Şaban Alaca, Kenneth S. Williams, ''Introductory algebraic number theory'',
Cambridge University Press Cambridge University Press is the university press of the University of Cambridge. Granted letters patent by Henry VIII of England, King Henry VIII in 1534, it is the oldest university press in the world. It is also the King's Printer. Cambr ...
, 2004. * * * * * * * *


External links

*{{Commonscat-inline Algebraic number theory Field (mathematics)