Concept
A theory that reads solvability by radicals from the symmetry structure of a polynomial’s roots, originating in work Galois completed by age 20.
Understand it in one breath
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.
At a glance
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.
Key formula
Key moments
Abel — no general formula for the quintic
Niels Henrik Abel proved that a general fifth-degree equation cannot be solved by radicals, but the deeper structural reason was still missing.
Galois — equations and symmetry groups
By 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.
Liouville publishes Galois’s work
Liouville announced the importance of Galois’s manuscripts in 1843 and published major papers in 1846, bringing their structural insight to a wider mathematical community.
Wiles — Galois representations in Fermat’s proof
Galois representations helped connect elliptic curves and modular forms, a central route in the proof of Fermat’s Last Theorem.
Modern applications
Algebraic geometry, number theory including the proof of Fermat’s Last Theorem, coding theory, and quantum computing.
Beyond MathVoyage
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