Galois Groups and Field Families
Galois Groups of Quintic Fields
Determining Galois groups in degree five, where solvability by radicals first fails and the resolvent degrees grow.
Engineering / MathematicsGalois Groups and Field Families2 min readKV-MATH-0621
Degree five is where solvability by radicals fails in general, and where the resolvent method starts to cost real work. There are five transitive groups and they form a chain.
The five groups
| Group | Order | Solvable |
|---|---|---|
| Cyclic of order 5 | 5 | Yes |
| Dihedral of order 10 | 10 | Yes |
| Frobenius group of order 20 | 20 | Yes |
| Alternating group on 5 letters | 60 | No |
| Symmetric group on 5 letters | 120 | No |
The identification
Galois group of a quintic
- Compute the discriminantA square places the group inside the alternating group, leaving three candidates.
- Sample cycle typesThe presence of a transposition or a four-cycle rules out the smaller groups.
- Compute the sextic resolventA degree six resolvent whose rational root detects containment in the Frobenius group of order twenty.
- RefineFurther resolvents distinguish cyclic from dihedral within the solvable branch.
- Is the discriminant a square?
- Yes — group is in the alternating groupCyclic of order 5, dihedral of order 10, or alternating
- No — group is not in the alternating groupFrobenius group of order 20 or symmetric
- Does the sextic resolvent have a rational root?
- Yes — group is solvableOne of the three smaller groups
- No — group is alternating or symmetric
Cycle type evidence
Cost
Solvability by radicals
A quintic is solvable by radicals exactly when its Galois group is contained in the Frobenius group of order twenty. Since that is decidable by a single resolvent, solvability is computationally straightforward even though the general quintic is not solvable.
Source. Henri Cohen, A Course in Computational Algebraic Number Theory, Springer GTM 138 — 6.3.4. 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.