Galois Groups and Field Families
The Galois Group Computation Problem
What computing a Galois group means, why the splitting field cannot be constructed, and the transitive group classification that makes the problem finite.
Engineering / MathematicsGalois Groups and Field Families2 min readKV-MATH-0618
The Galois group of a polynomial is the automorphism group of its splitting field, realised as a permutation group on the roots. Computing it means identifying which transitive group it is, not constructing the splitting field.
The problem
Given an irreducible polynomial of degree n over the rationals, determine its Galois group as a subgroup of the symmetric group on n letters, up to conjugacy.
Why the problem is finite
The group must be a transitive subgroup of the symmetric group, since the polynomial is irreducible. Transitive subgroups have been classified for all small degrees, giving a finite list of candidates.
| Degree | Number of transitive groups |
|---|---|
| 3 | 2 |
| 4 | 5 |
| 5 | 5 |
| 6 | 16 |
| 7 | 7 |
| 8 | 50 |
The two main invariants
The discriminant
A square discriminant means the group lies in the alternating group. This single test halves the candidate list immediately and costs one discriminant computation.
Factorisation shapes modulo primes
The degrees of the irreducible factors modulo an unramified prime give the cycle type of a Frobenius element. Sampling many primes reveals which cycle types occur.
Cycle types and Chebotarev
By the Chebotarev density theorem, each conjugacy class of the group occurs as a Frobenius cycle type with density proportional to its size. Sampling factorisation shapes therefore samples the group's conjugacy classes.
The resolvent approach
To decide conclusively, resolvent polynomials are used: their factorisation patterns distinguish candidate groups exactly. See the resolvent method.
Source. Henri Cohen, A Course in Computational Algebraic Number Theory, Springer GTM 138 — 6.3. 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.