Let all varieties over an algebraically closed field $k$ of arbitrary characteristic.
Let $f:V\to B$ be a surjective morphism where $V$ is a smooth surface and $B$ is a smooth complete curve.
I've read here (2.2, page 82) that if $F$ is a singular fiber of $f$, then $F$ is connected. In the proof, the author argues that this is a consequence of "Zariski's connectedness theorem".
The only reference I could find was in this wikipedia article, in a statement which asks for the function field of $B$ to be "separably closed" in the function field of $V$.
I'm not sure what "separably closed" means here and the author in the first link didn't mention anything about it.
What's going on?
(by the way, I imagine that Zariski's original statement is here in chapter 6, but I couldn't have access to it)