Algebra Series of 6 60 min
Abel–Ruffini
Why no formula in radicals solves every quintic: Arnold's short proof by loops and commutators, then the Galois-theory version that names specific unsolvable quintics.
What you need Vol. I follows Arnold's 1963 lectures to Moscow high-school students: complex numbers, loops and permutations, with commutators built on the page.
Start with Vol. I 2026
- I The Short Proof Arnold's topological proof that no radical formula solves the general quintic, using loops of coefficients, winding numbers and nested commutators in S₅.
- II Roots Exist The fundamental theorem of algebra, proved by a minimum modulus argument, and how complex conjugation permutes the five roots of a real quintic.
- III Symmetries of the Roots Permutations of the roots, parity, normal subgroups, a proof that A₅ is simple, and why S₄ is solvable while S₅ is not.
- IV Where the Roots Live Splitting fields, radical towers, and the Galois group as the root permutations that field automorphisms realize, worked out for the cube root of 2.
- V Towers and the Galois Correspondence The tower law, Kummer theory and the Galois correspondence, and how they show a polynomial is solvable by radicals exactly when its Galois group is solvable.
- VI Quintic Symmetry Why the general polynomial of degree n has Galois group Sₙ, a quintic over ℚ with group S₅, a solvable quintic, and the history from Ruffini to Galois.