Linear patterns of prime elements in number fields

We prove a number field analogue of the Green–Tao–Ziegler theorem on simultaneous prime values of degree 1 polynomials whose linear parts are pairwise linearly independent. Applications of our results include a Hasse principle of rational points for certain fibrations $X\to {\mathbb{P}}^1$ over number fields $K$ which had only been available over $\mathbb{Q}$ by Harpaz–Skorobogatov–Wittenberg, and construction of elliptic curves having some specified ranks due to Koymans–Pagano and Zywina. This latter family of results led to a negative answer to a generalized Hilbert Tenth Problem.