Mathematics - Abstract Algebra - Galois theory - Why Some Equations Can't Be Solved

12 steps: (1) Quadratic formula solves all degree-2 polynomials. Cubic and quartic formulas exist too (complex!). (2) QUESTION: Is there a formula for degree 5 and above? (3) Abel-Ruffini theorem: NO general formula using radicals for degree ≥5! (4) GALOIS' insight: connect solvability to GROUP SYMM