  We construct examples of $K3$ surfaces of geometric Picard rank $1$. Our
method is a refinement of that of R. van Luijk. It is based on an analysis of
the Galois module structure on \'etale cohomology. This allows to abandon the
original limitation to cases of Picard rank $2$ after reduction modulo $p$.
Furthermore, the use of Galois data enables us to construct examples which
require significantly less computation time.
