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.
- 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. Its discriminant is 148, the smallest discriminant of a non-cyclic totally real cubic field.
- 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).
Read more about this topic: Cubic Field
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