Past exam of the mathematics course of the University of Cambridge / 2025 / ib / Paper 3 / 10E / a / i / Solution

If a prime divided every coefficient of $fg$, reducing modulo that prime would make the product of the two nonzero reduced primitive polynomials zero in a domain, impossible. Thus a product of primitive polynomials is primitive. Factoring the contents from arbitrary $f,g$ then givesfor a unit $u$.