Galois Groups and Field Families
Galois Groups of Cubic and Quartic Fields
Complete determination of Galois groups in degrees three and four, where the discriminant and one cubic resolvent decide everything.
Engineering / MathematicsGalois Groups and Field Families2 min readKV-MATH-0620
Degrees three and four are small enough to be settled completely by hand. They are the standard worked examples and are worth knowing because they show the general method in a case where every step is explicit.
Degree three
There are two transitive subgroups of the symmetric group on three letters, and the discriminant distinguishes them.
| Discriminant | Galois group | Order |
|---|---|---|
| A perfect square | Cyclic of order 3 | 3 |
| Not a square | Symmetric group on 3 letters | 6 |
Degree four
There are five transitive subgroups. The discriminant and the factorisation of the resolvent cubic together distinguish all of them.
| Resolvent cubic factorisation | Discriminant | Galois group |
|---|---|---|
| Irreducible | Not a square | Symmetric group on 4 letters |
| Irreducible | A square | Alternating group on 4 letters |
| Splits completely | A square | Klein four group |
| One rational root only | Not a square | Dihedral of order 8, or cyclic of order 4 |
Why the resolvent cubic exists
The symmetric group on four letters has a normal subgroup — the Klein four group — with quotient the symmetric group on three letters. That quotient map is realised by the resolvent cubic, which is why degree four reduces to degree three.
Field families
Cubic fields with cyclic Galois group are exactly the cyclic cubic fields, which form a well-understood family — see cubic field families.
Source. Henri Cohen, A Course in Computational Algebraic Number Theory, Springer GTM 138 — 6.3.2-6.3.3. 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.