Assess students' ability to determine the Galois group over Q of an irreducible quintic by applying irreducibility tests, computing the discriminant, using factorization patterns modulo primes to infer cycle types, and constructing/using appropriate resolvent polynomials; require narrowing the group to one of the transitive subgroups of S5 (S5, A5, D5, C5, Frobenius F20, etc.) and deciding whether the polynomial is solvable by radicals with justification. Specify which computations and group-theoretic deductions are expected.