  We prove that the von Mangoldt function of a number field $K$ is well
approximated by its Cramer/Siegel models in the Gowers norm sense. Via the
inverse theory of Gowers norms, this is achieved by showing that the difference
of the von Mangoldt function and its model is asymptotically orthogonal to
nilsequences. To prove the asymptotic orthogonality, we use Mitsui's Prime
Number Theorem as the base case and proceed by upgrading Green--Tao's type I/II
computation to the general number field. As an application, we prove a number
field analog of the Green--Tao--Ziegler theorem on simultaneous prime values of
affine-linear forms of finite complexity. This in turn can be used to prove a
Hasse principle result for certain fibrations $X\to \mathbb{P}^1$ over a number
field $K$ extending a result of Harpaz--Skorobogatov--Wittenberg which was only
available over $\mathbb Q$.
