  A central question in Arithmetic geometry is to determine for which
polynomials $f \in \mathbb{Z}[t]$ and which number fields $K$ the Hasse
principle holds for the affine equation $f(t) =
N_{K/\mathbb{Q}}(\boldsymbol{x}) \neq 0$. Whilst extensively studied in the
literature, current results are largely limited to polynomials and number
fields of low degree. In this paper, we establish the Hasse principle for a
wide family of polynomials and number fields, including polynomials that are
products of arbitrarily many linear, quadratic or cubic factors. The proof
generalises an argument of Irving, which makes use of the beta sieve of Rosser
and Iwaniec. As a further application of our sieve results, we prove new cases
of a conjecture of Harpaz and Wittenberg on locally split values of polynomials
over number fields, and discuss consequences for rational points in fibrations.
