Galois Groups and Field Families
Constructing Tables of Number Fields
Systematically enumerating number fields of small degree and bounded discriminant, with canonical representatives and completeness arguments.
Engineering / MathematicsGalois Groups and Field Families2 min readKV-MATH-0624
Tables of number fields ordered by discriminant are a standard research resource. Constructing one requires enumerating candidates, removing duplicates, and arguing that nothing was missed.
The task
List all number fields of a given degree and signature whose discriminant is below a bound, each represented by a canonical defining polynomial, with a proof that the list is complete.
The search
Every field contains an algebraic integer generating it whose conjugates are bounded in terms of the discriminant. Enumerating candidate minimal polynomials within those bounds and testing each produces all fields.
Constructing a table of number fields
- Bound the coefficientsHunter-style bounds limit the coefficients of a minimal polynomial of a suitable generator.
- EnumerateLoop over integer coefficient tuples within the bounds.
- FilterDiscard reducible polynomials and those with discriminant above the target bound.
- Compute invariantsMaximal order, field discriminant, signature.
- CanonicaliseReduce to a canonical polynomial — see polynomial reduction.
- DeduplicateIdentical canonical polynomials mean identical fields.
The completeness argument
Deduplication
Cost
| Stage | Cost driver |
|---|---|
| Enumeration | Grows quickly with degree and discriminant bound |
| Irreducibility testing | Cheap; done first as a filter |
| Maximal order computation | Per candidate; dominates for large searches |
| Deduplication | Isomorphism tests between same-discriminant fields |
Families
Some families admit direct parametrisation, avoiding the search entirely — see cubic field families. Where a parametrisation exists it is vastly more efficient and gives completeness by construction.
Published tables
Extensive tables exist for small degrees and are the practical starting point for most work — see published tables.
Source. Henri Cohen, A Course in Computational Algebraic Number Theory, Springer GTM 138 — 6.4.1. 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.