Departure question
Move sameness aroundPort 2 of 8Use rotations, permutations, matrices, and categories to read transformation rules and relationships rather than appearances.
This voyage is an editorial path for understanding, not a claim of direct historical influence or sole invention.
The decisive move is to translate “is this equation solvable by radicals?” into “is its Galois group solvable?” General radical formulas exist for quadratic, cubic, and quartic equations, but not for the general quintic. Galois grouped the permutations of roots that preserve their algebraic relations and opened the route to the solvable-group criterion. His testamentary letter to Auguste Chevalier summarized work already in progress; Liouville announced the manuscripts’ importance in 1843 and published major papers in 1846.
Degree n | Galois group | Formula by radicals | Solvable group? |
|---|---|---|---|
2 (quadratic) | S₂ ≅ ℤ/2 | √discriminant | ✓ |
3 (cubic) | S₃ | Cardano formula | ✓ (solvable) |
4 (quartic) | S₄ | Ferrari formula | ✓ (solvable) |
General quintic | S₅ (contains the simple group A₅) | No general formula | ✗ |
S₅ is not solvable, so the general quintic has no formula by radicals. Special quintics whose Galois groups are solvable can still be solved by radicals.
A theory that reads solvability by radicals from the symmetry structure of a polynomial’s roots, originating in work Galois completed by age 20.
Ports in time
Follow the scenes to see problems, notation, standards of proof, and applications changing across different times and places.
Niels Henrik Abel proved that a general fifth-degree equation cannot be solved by radicals, but the deeper structural reason was still missing.
No reliable place is given, so time continues without an invented pin
Continue through the world of this yearBy age 20, Galois had connected permutations of roots with solvability by radicals. His approximately seven-page testamentary letter summarized work already in progress rather than creating the whole theory overnight.
The recorded place matches a canonical map anchor
Continue this scene on the mapLiouville announced the importance of Galois’s manuscripts in 1843 and published major papers in 1846, bringing their structural insight to a wider mathematical community.
No reliable place is given, so time continues without an invented pin
Continue through the world of this yearGalois representations helped connect elliptic curves and modular forms, a central route in the proof of Fermat’s Last Theorem.
No reliable place is given, so time continues without an invented pin
Continue through the world of this yearAlgebraic geometry, number theory including the proof of Fermat’s Last Theorem, coding theory, and quantum computing.
Curated sources and problems. Bring one discovery back from OEIS, Project Euler, MathOverflow, or arXiv.
No concept belongs to one person
These are not inventor credits. They are different ports: opening a problem, sharpening a language, or carrying it into another world.
Number lenses
These numbers are editorial lenses for the voyage, not required prerequisites.
Use rotations, permutations, matrices, and categories to read transformation rules and relationships rather than appearances.
Open the number voyage
Use rotations, permutations, matrices, and categories to read transformation rules and relationships rather than appearances.
Open the number voyage
Use rotations, permutations, matrices, and categories to read transformation rules and relationships rather than appearances.
Open the number voyage
Concept genealogy
Concepts arriving from before
Current port
Galois Theory
Concepts opened from here
No direct successor port is curated yet.
Only direct editorial links are shown; this is not a complete learning order or historical influence line.
