Galois Groups and Field Families
Galois Groups of Sextic and Septic Fields
Degrees six and seven, where the number of candidate groups and the resolvent degrees make careful strategy essential.
Engineering / MathematicsGalois Groups and Field Families2 min readKV-MATH-0622
Degree six has sixteen transitive groups and degree seven has seven. Six is the harder case despite the lower degree, because six is composite and admits imprimitive groups with block structure.
Degree seven
Seven is prime, so every transitive group is primitive and the list is short. The groups form a chain ordered by inclusion, which makes identification a sequence of containment tests.
| Group | Order | Note |
|---|---|---|
| Cyclic of order 7 | 7 | Smallest |
| Dihedral of order 14 | 14 | |
| Metacyclic of order 21 | 21 | |
| Frobenius group of order 42 | 42 | Largest solvable |
| Simple group of order 168 | 168 | |
| Alternating group on 7 letters | 2520 | |
| Symmetric group on 7 letters | 5040 |
Degree six
Six is composite, so a degree six field may contain a quadratic or cubic subfield. The corresponding groups are imprimitive: they preserve a partition of the roots into blocks.
- Primitive group
- Preserves no non-trivial partition. The field has no proper subfield other than the rationals.
- Imprimitive group
- Preserves a partition into blocks. Corresponds to a proper subfield, of degree two or three.
- Block system
- The partition preserved. Finding one identifies a subfield — see the subfield problem.
Strategy
Strategy for degree six
- Test the discriminantSquare or not.
- Find subfieldsRestricts to imprimitive groups if any exist — see subfields.
- Sample cycle typesEliminates further candidates cheaply.
- Apply targeted resolventsOnly for the candidates still standing.
Higher degrees
Verification
Whatever method is used, the identification should be confirmed against test polynomials with known groups — see test polynomials.
Source. Henri Cohen, A Course in Computational Algebraic Number Theory, Springer GTM 138 — 6.3.5-6.3.6. 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.