  Let $K=\mathbb{Q}(\omega)$ with $\omega$ the root of a degree $n$ monic
irreducible polynomial $f\in\mathbb{Z}[X]$. We show the degree $n$ polynomial
$N(\sum_{i=1}^{n-k}x_i\omega^{i-1})$ in $n-k$ variables formed by setting the
final $k$ coefficients to 0 takes the expected asymptotic number of prime
values if $n\ge 4k$. In the special case $K=\mathbb{Q}(\sqrt[n]{\theta})$, we
show $N(\sum_{i=1}^{n-k}x_i\sqrt[n]{\theta^{i-1}})$ takes infinitely many prime
values provided $n\ge 22k/7$.
  Our proof relies on using suitable `Type I' and `Type II' estimates in
Harman's sieve, which are established in a similar overall manner to the
previous work of Friedlander and Iwaniec on prime values of $X^2+Y^4$ and of
Heath-Brown on $X^3+2Y^3$. Our proof ultimately relies on employing explicit
elementary estimates from the geometry of numbers and algebraic geometry to
control the number of highly skewed lattices appearing in our final estimates.
