Galois Groups and Field Families
Cyclic and Pure Cubic Field Families
Cyclic cubic and pure cubic fields as parametrised families, with closed-form invariants that avoid general algorithms.
Engineering / MathematicsGalois Groups and Field Families2 min readKV-MATH-0625
Some families of number fields are parametrised explicitly, so their invariants follow from formulas rather than from general algorithms. Cubic fields provide the two standard examples and they are instructive opposites.
Cyclic cubic fields
A cyclic cubic field is Galois over the rationals with cyclic group of order three. By the Kronecker-Weber theorem it lies inside a cyclotomic field, which gives a complete parametrisation.
| Property | Value |
|---|---|
| Galois group | Cyclic of order 3 |
| Discriminant | The square of the conductor |
| Conductor | A product of primes congruent to 1 modulo 3, possibly times 9 |
| Signature | Totally real; r1 = 3, r2 = 0 |
| Unit rank | 2 |
Pure cubic fields
A pure cubic field is generated by a cube root of a rational integer. It is never Galois, having only one real embedding among three.
| Property | Value |
|---|---|
| Galois group | Symmetric group on 3 letters |
| Signature | r1 = 1, r2 = 1 |
| Unit rank | 1 |
| Discriminant | Given by a closed formula with a case split |
| Normal closure | Degree six, obtained by adjoining a cube root of unity |
Why parametrised families are useful
Enumeration without search
The family is generated directly from a parameter, so tables are complete by construction rather than by a bounding argument.
Closed-form invariants
Discriminant, signature and integral basis follow from formulas, avoiding Round 2 entirely.
Test material
Fields with known invariants make excellent regression tests for general algorithms.
The contrast
Other families
Similar parametrisations exist for cyclic quartic fields, biquadratic fields, and cyclotomic fields and their subfields. In each case class field theory supplies the parametrisation, and the general algorithms serve only as verification.
Source. Henri Cohen, A Course in Computational Algebraic Number Theory, Springer GTM 138 — 6.4.2-6.4.3. Structural reference unverified: the source file was not available during authoring; chapter and section numbers are taken from the published edition and have not been checked against a physical copy.