The Insolvability of the Quintic

Radical extension towers produce corresponding Galois group towers that shrink at each level. This connection between field extensions and subgroup chains reveals the algebraic structure underlying solvability by radicals.

A polynomial is solvable by radicals if its roots can be expressed using arithmetic operations and nth roots alone. This formalizes when formula-based solutions like the quadratic formula exist for polynomial equations.

A group is solvable if it admits a subgroup chain where consecutive quotients have prime order. This group-theoretic property precisely characterizes when polynomials are solvable by radicals through Galois theory.

The symmetric group Sₙ consists of all permutations of n elements. Its solvability determines whether general degree-n polynomials admit radical formulas. For n ≥ 5, Sₙ is non-solvable, proving no general quintic formula exists.