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Concept
Cubic field
Summary
In mathematics, specifically the area of algebraic number theory, a cubic field is an algebraic number field of degree three.
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.
A cubic field is called a pure cubic field if it can be obtained by adjoining the real cube root 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 non-real cube roots.
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.
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.
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.
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.
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