Computing Galois Groups of Number Fields

Executive summary

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

Transitive subgroups 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 discriminant does half the work in low degree

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.

AlgorithmNarrowing by cycle typesin: T → out: a reduced list of candidate Galois groups

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.

Statistics cannot prove absence

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.

AlgorithmThe resolvent testin: T, candidate subgroup H → out: whether Gal(T) ⊆ H up to conjugacy

Choose an invariant F of the target subgroup H.
Form the resolvent R(x) = ∏(x − F^σ) 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.

Precision or exactness

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.

Degree is the hard limit

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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ProvenanceSources and further reading

This page is an original KEVOS explanatory article. It presents the underlying mathematics — definitions, algorithms, complexity results and selection criteria — in KEVOS editorial voice. No text is reproduced from any copyrighted source. Where numerical tables are relevant, KEVOS links to live authoritative databases rather than republishing static values.

Handbook application: from concept to controlled practice

Purpose. This expanded section turns the original page into a practical handbook. It preserves the supplied material and adds a repeatable way to apply, check and review Computing Galois Groups of Number Fields. It does not replace a contract, legislation, a controlled standard, competent engineering judgement or specialist advice.

The operating aim is to turn a compact mathematical statement into a usable chain of definitions, claims, examples and checks. Read the original explanation first, then use the workflow and checks below to convert knowledge into evidence.

Treat Computing Galois Groups of Number Fields as a network of definitions and implications, not as a list of formulas. The working vocabulary on this page—section, galois, groups, number, resolvent—should be made explicit before any proof or computation begins. Record the ambient set or structure, the permitted operations and the equality or equivalence relation in use. A compact theorem often changes meaning when the base field, finiteness condition, commutativity assumption or direction of an action changes.

For a proof, write the hypotheses as a checklist and mark the line at which each one is used. For a computation, state the representation of the input, the arithmetic model, the termination condition and the output invariant. For a classification problem, distinguish existence from uniqueness and distinguish an object from its representation. These separations prevent a correct local calculation from being mistaken for the general result.

A useful worked example should be small enough to inspect completely but rich enough to exercise the main mechanism. Compute the result in two ways where practical: symbolically and by substitution, structurally and numerically, or directly and through a normal form. Then include one near-miss example in which a hypothesis fails. The contrast explains why the theorem is shaped as it is and gives the reader a diagnostic pattern for later problems.

Verification is part of the mathematics. Check domains and codomains, substitute proposed solutions, test identity and zero cases, compare dimensions or cardinalities, and confirm that maps respect the required operations. In numerical work, report precision, conditioning and a residual rather than digits alone. In algorithmic work, separate mathematical correctness from implementation complexity and resource limits.

Step-by-step operating method

Fix the setting. State the objects, ambient structure, notation and assumptions before manipulating symbols.
Separate claims. Distinguish definitions, hypotheses, conclusions, equivalent conditions and consequences.
Choose a method. Select proof, construction, calculation or algorithm according to the question actually asked.
Work a small case. Use the smallest non-trivial example to expose the mechanism and test edge behaviour.
Verify independently. Substitute back, check invariants, test boundary cases or use an alternative derivation.

Worked-example protocol

Illustrative method—not a source theorem. Start with a small admissible input and list the definitions it must satisfy. Carry out each transformation on a separate line, citing the property that permits it. Preserve exact values until approximation is necessary. At the end, verify the output against the original definition and one invariant such as dimension, degree, determinant, order, norm or residual. Then alter one hypothesis and observe which step ceases to be valid. This protocol creates a reusable example without inventing a theorem-specific numerical answer.

Stage	Record	Quality check
Input	Objects, domain, notation, assumptions	Every symbol is defined
Method	Permitted operation or cited result at each step	All hypotheses hold
Output	Exact result and representation	Correct type, domain and form
Verification	Substitution, invariant or alternative derivation	Independent agreement
Boundary test	Zero, identity, degenerate or failed hypothesis	Scope is understood

Common failure modes and recovery actions

1. Watch for

Using a theorem without checking every hypothesis.

Recovery: Return to the governing definition or requirement and restate the decision in one sentence.

2. Watch for

Treating a suggestive example as a proof of the general case.

Recovery: Separate evidence from assumption, assign an owner and set a date for validation.

3. Watch for

Changing notation or conventions part-way through an argument.

Recovery: Run a small counterexample, boundary test, pilot or independent check before proceeding.

4. Watch for

Hiding a division-by-zero, convergence, finiteness or commutativity assumption.

Recovery: Record the consequence, decision and rationale, then update the controlled baseline.

5. Watch for

Reporting a computed result without a residual, substitution or structural check.

Recovery: Escalate when the issue affects safety, compliance, acceptance, material value or an agreed tolerance.

Review checklist

Can every symbol be traced to a definition or prior result?
Which hypothesis does each major step use?
Does the method cover zero, identity, degenerate and boundary cases?
Can the conclusion be checked by a second representation or calculation?
Are mandatory requirements distinguished from recommendations and illustrative values?
Are sources, assumptions, units, dates and versions recorded closely enough to reproduce the decision?
Have safety, legal, ethical, stakeholder and operational consequences been considered at the appropriate level?
Is there a named owner and a trigger for review, escalation, change or retirement?

Questions for deeper application

What is the most important distinction a practitioner must preserve when applying Computing Galois Groups of Number Fields?