Given a field $K$, $K$-schemes $X$ and $Y$, an étale map $f:X\to Y$, $x\in X$ a point of the topological space to $X$ and $y:=f(x)\in Y$ a point of the topological space to $Y$. What is the basechange $X\times_Y Spec(k(y))$ of $f$ along $g:Spec(k(y))\to Y$?
I think it is a finite disjoint union $\coprod_n Spec(K_n)$ for fields $K_n$, all (algebraic?) extensions of $K$. The union is indexed by the number of preimages of $y$ for the topological space map $f$. Is this true?
If $g$ is a geometric point ($k(y)=\bar K$) of $Y$, is it true that I can find $g':Spec(k(y))\to X$ with $g=f\circ g'$?