Let  be a polynomial with integer coefficients of degree less than or equal to 7, and assume f to be reducible over the rational numbers. We show how to compute the Galois group of f. The main tools we employ are composita of irreducible factors, discriminants, and in the case when f is a degree 7 polynomial, two linear resolvents. For each possible Galois group G, we provide a reducible polynomial whose Galois group over the rationals is G.

reducible polynomials, Galois groups, discriminants, linear resolvents.