Galois Groups and Field Families
The Resolvent Method for Galois Groups
Constructing resolvent polynomials whose factorisation distinguishes candidate Galois groups, and the practical issues in using them.
Engineering / MathematicsGalois Groups and Field Families2 min readKV-MATH-0619
A resolvent is a polynomial built from the roots whose factorisation pattern depends on the Galois group. Choosing resolvents that separate the candidate groups turns group identification into polynomial factorisation.
The construction
Choose a polynomial function of the roots that is invariant under a candidate subgroup. The resolvent is the product over coset representatives of the linear factors built from the images of that function.
Reading the result
The resolvent has a rational root exactly when the Galois group is contained in a conjugate of the candidate subgroup. More generally, the degrees of its irreducible factors identify the group among the candidates.
Galois group by resolvents
- Compute the discriminantSquare or not — halves the candidate list.
- Sample cycle typesFactor modulo several primes to eliminate further candidates.
- Choose a resolventOne that separates the remaining possibilities.
- Compute itBy resultants or symmetric functions.
- Factor itOver the rationals — see integer factorisation.
- IdentifyRead the group from the factorisation pattern.
The repeated root problem
Cost
| Stage | Cost driver |
|---|---|
| Discriminant | One resultant |
| Cycle type sampling | Cheap; factorisation modulo small primes |
| Resolvent construction | Degree equal to the index of the subgroup; grows quickly |
| Resolvent factorisation | The dominant cost; large degree with large coefficients |
Numerical shortcuts
Rather than computing a resolvent exactly, its roots can be approximated from numerical roots of the original polynomial and a rational root recognised directly. This is much faster but requires exact verification, exactly as in dependence detection.
Source. Henri Cohen, A Course in Computational Algebraic Number Theory, Springer GTM 138 — 6.3.1. 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.