Narrow by factorisation statistics, confirm by resolvents
The Galois group of an irreducible degree-n polynomial is a transitive subgroup of the symmetric group on n letters, and there are few of these for small n. Factoring the polynomial modulo many primes gives cycle types that must occur in the group, which narrows the candidates quickly by Chebotarev. Resolvent polynomials then decide definitively: a resolvent has a rational root exactly when the group lies in a specified subgroup.
Learning objectives
- List the transitive groups for small degree and their inclusion lattice.
- Use Dedekind's theorem to obtain cycle types from factorisations modulo p.
- Construct and test a resolvent polynomial.
- Combine the two methods into a decision procedure.
- Recognise the difficulty of the general high-degree case.
Section 01Transitive groups by degree
| Degree | Number of transitive groups | The groups |
|---|---|---|
| 2 | 1 | C2 |
| 3 | 2 | C3, S3 |
| 4 | 5 | C4, V4, D4, A4, S4 |
| 5 | 5 | C5, D5, F20, A5, S5 |
| 6 | 16 | Including C6, S3, D6, A4, PGL2(5), A6, S6 |
| 7 | 7 | C7, D7, F21, F42, PSL3(2), A7, S7 |
The Galois group lies in the alternating group exactly when the discriminant is a perfect square. In degree 3 this single test distinguishes C3 from S3 completely; in degree 4 it halves the candidate list at once.
Section 02Cycle types from factorisation
Dedekind's theorem states that for a prime p not dividing the discriminant, the degrees of the irreducible factors of T modulo p give the cycle type of a Frobenius element in the Galois group.
- For many primes p not dividing disc(T), factor T modulo p.
- Record the multiset of factor degrees as a cycle type. Each observed type must occur in the Galois group.
- Eliminate every candidate group not containing an element of that cycle type.
- By Chebotarev, cycle types appear with frequency proportional to their share of the group, so all types appear after enough primes.
- Stop when one candidate remains, or when the remaining candidates share all cycle types and a resolvent is required.
Not observing a cycle type after many primes is evidence, not proof, that the group lacks it. Chebotarev gives densities, not guarantees, so a conclusion reached by elimination alone is heuristic and must be confirmed by an exact method.
Section 03Resolvent polynomials
For a subgroup H of the symmetric group, choose a polynomial in the roots that is invariant precisely under H. Its orbit under the full group yields a resolvent whose coefficients are rational, and which has a rational root exactly when the Galois group is contained in a conjugate of H.
- Choose an invariant F of the target subgroup H.
- Form the resolvent R(x) = ∏(x − Fσ) over coset representatives σ, with coefficients computed symbolically from T or numerically from the roots and rounded.
- Test whether R has a rational root. A root means the Galois group is contained in a conjugate of H.
- If R has repeated roots, perform a Tschirnhaus transformation on T and restart — repeated roots make the test inconclusive.
- Descend through the subgroup lattice, testing each level, until the group is pinned down.
Resolvent coefficients can be computed exactly by symmetric function manipulation, or numerically from the complex roots and rounded. The numerical route is far faster but requires enough precision that rounding is certain — the same discipline as elsewhere in the subject.
Section 04The combined procedure and its limits
- Stage 01Test the discriminantA square discriminant places the group inside the alternating group.
- Stage 02Collect cycle typesFactor modulo many primes; eliminate candidates lacking observed types.
- Stage 03Apply resolventsResolve the remaining ambiguity exactly by descending the subgroup lattice.
- Stage 04VerifyConfirm the answer is consistent with every observed cycle type and with the discriminant test.
The number of transitive groups grows rapidly — there are already thousands by degree 16 — and suitable invariants become harder to construct and their resolvents harder to compute. Beyond moderate degree, general Galois group determination remains genuinely difficult, and specialised methods exploiting known structure are used instead.
ReferenceFrequently asked questions
Why is the Galois group of a number field worth computing?
Because it determines the subfield lattice, the splitting behaviour of primes, and whether the extension is abelian — and hence whether class field theory applies. Many structural questions reduce to knowing the group.
What is a Tschirnhaus transformation for?
It replaces the defining polynomial by another with the same splitting field but different roots, which breaks the accidental coincidences causing repeated resolvent roots. It is the standard repair when a resolvent test is inconclusive.
Can the Galois group be read from the factorisation pattern alone?
Only when the candidate groups have distinct cycle type sets. For degrees up to about 7 this resolves most cases, but pairs such as C4 and V4 require a resolvent, so the exact method cannot be dispensed with.
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