Assess the ability to apply Galois theory to determine whether polynomials (typically over Q or a given base field) are solvable by radicals. Tasks include computing splitting fields and Galois groups, analyzing group structure for solvability via composition series with abelian quotients, using discriminant/resolvent criteria and known group classifications (e.g. S5, A5), and constructing or ruling out radical expressions for explicit cubic, quartic, and irreducible quintic examples; require clear justification of each conclusion.