Definition 17.11.1 (Radical extension). A field extension is a radical extension if there is a tower where each with for some . A polynomial is solvable by radicals if its splitting field is contained in some radical extension of .
Theorem 17.11.1 (Galois’s criterion). A polynomial is solvable by radicals if and only if its Galois group (where is the splitting field of ) is a solvable group.
Note. Recall that a group is solvable if it has a subnormal series with each abelian. For , the group is solvable (e.g., where ). The key fact: is simple (proved in Lecture 11), so is not solvable.
Corollary 17.11.1 (Insolvability of the general quintic). There is no formula involving only the four arithmetic operations and radicals that expresses the roots of a general degree-5 polynomial in terms of its coefficients. More precisely: there exist quintic polynomials whose Galois group is , which is not solvable.
Example 17.11.1. The polynomial is irreducible (Eisenstein at ) and has Galois group . It has exactly three real roots and one pair of complex conjugate roots: the complex conjugation gives a transposition in , and irreducibility gives a 5-cycle. A transposition and a 5-cycle generate .
Note. This is the Abel–Ruffini theorem (1824), with the group-theoretic formulation due to Galois (1832). For degree , the classical formulas (quadratic, Cardano, Ferrari) exist precisely because are solvable. Galois theory unifies and explains all of this.