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
:
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