Galois Groups and Field Families
Test Polynomials for Galois Group Software
Why Galois group implementations need a curated test set, what a good set covers, and how to build one.
Engineering / MathematicsGalois Groups and Field Families2 min readKV-MATH-0623
Galois group computation is error-prone and its errors are quiet: a misidentified group is a plausible-looking answer. A curated set of polynomials with known groups is the only practical defence.
What a good test set covers
| Category | Why it matters |
|---|---|
| One polynomial per transitive group, each degree | Basic coverage; catches gross misidentification |
| Polynomials whose resolvents have repeated roots | Exercises the Tschirnhaus fallback path |
| Large coefficients | Exercises coefficient growth handling |
| Small discriminants | Fields with many ramified primes |
| Imprimitive groups in composite degree | Exercises subfield detection |
| Near-miss pairs | Groups distinguished only by a specific resolvent |
Constructing test polynomials
Building test cases with known groups
- Start from a known groupConstruct a field with the desired group, for instance as a subfield of a cyclotomic field.
- Find a defining polynomialCompute the minimal polynomial of a generator.
- ReduceApply polynomial reduction to obtain small coefficients.
- RecordStore the polynomial with its verified group, discriminant and any subfields.
Cross-checking
Beyond a fixed test set, results should satisfy structural constraints, and checking these catches errors on inputs not in the set.
- The group order must be divisible by the field degree, since the group is transitive.
- The group must lie in the alternating group exactly when the discriminant is a square.
- Observed cycle types must all occur in the identified group.
- The subfield lattice must match the subgroup lattice.
Regression discipline
Source. Henri Cohen, A Course in Computational Algebraic Number Theory, Springer GTM 138 — 6.3.7. 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.