Covers of varieties with completely split points

Let K be a field such that all Sylow subgroups of its absolute Galois group G(K) are infinite. Let X be a smooth variety over K with function field F and Y -> X the normalisation in a finite, separable extension E vertical bar F. We show: If there is a closed point x is an element of X which does not split completely in Y -> X, then the set of these points is Zariski dense in X.