Assess students' ability to determine whether a given irreducible quintic is solvable by radicals using Galois theory. Questions will require computing or deducing Galois groups of quintic polynomials (via discriminant, resolvents, mod‑p reduction and cycle types), describing splitting fields and intermediate extensions, applying group‑theoretic solvability criteria (recognizing S5, A5 and solvable subgroups), constructing explicit radical solutions when possible, and proving unsolvability for standard examples.