Definition
If K is a field extension of the rational numbers Q 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 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, 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 field Q.
Read more about this topic: Cubic Field
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