# Is the other extreme of Hilbert Irreducibility true?

Let $K$ be a number field (or perhaps more generally a Hilbertian field). Let $X_K\rightarrow \mathbb{P}^1_K$ be a regular (i.e. without extension of scalars) $G$-Galois branched cover. Hilbert's Irreducibility implies that there are infinitely many $K$-rational points on $\mathbb{P}^1_K$ such that the fiber product $Spec(K)\times_{\mathbb{P}^1_K}X_K$ is connected. (This is frequently stated as: there are infinitely many $K$-rational points on $\mathbb{P}^1_K$ such that specializing to them gives a $G$-Galois extension of fields over $K$.)

My question is whether the other extreme of this is true. I.e., is there a $K$-rational point on $\mathbb{P}^1_K$ such that $Spec(K)\times_{\mathbb{P}^1_K}X_K$ has $|G|$ connected components (each one isomorphic to $Spec(K)$)? In other words, is there a $K$-rational point on $\mathbb{P}^1_K$ that "splits completely"?

This reminds me of the statement that in a Galois extension of number fields there are infinitely many primes that split. This is a far from trivial statement that comes from Class Field Theory. However the analogy isn't perfect, so I don't immediately see how the same methods can be used here.

-
I am reasonably certain that the statement that there are infinitely many split primes is actually quite elementary; the argument in mathoverflow.net/questions/15220/… should extend. –  Qiaochu Yuan Jan 15 '12 at 2:30