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Let $X$ be a scheme over base field $K$ and denote by $R[X](= \Gamma(X,\mathcal{O}_X))$ the ring of regular global functions on $X$.

Question: Assume that $k$ is algebraically (resp. separably) closed in $R[X]$, ie if $f \in k[X]$ monic polynomial and $a \in R[X]$ with $f(a)=0$, then $a \in k$ (resp. $f$ not separable).
What can we basically draw from this about topological/geometrical properties of $X$?

Remarks: One can ask a slightly diffrent variant of this question (which is not a generalization of it but of similar flavour): If $f:X \to Y$ a morphism of integral schemes, consider the extension $F(X)/F(Y)$ of their function fields.

Then, if $K(Y)$ is algebraically (resp. separably) closed in $K(X)$, which conclusions can be drawn about geometry of th3 map $f$?

One special case it easy: If $X$ is proper, then using Stein decomposition, we see that $R[X]$ is finite over $k$, especially algebraic, so if $k$ algebraically closed in $R[X]$ then $R[X]=k$ and $X$ connected. The same if $k$ separably closed, eg finite or of characteristic zero.

Is this reasoning correct? Can we say more about geometry of $X$, resp the geometry of $f$ in relativ3 situation.

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    $\begingroup$ Have you considered $f(X)=X^2-X$ and what this tells you? $\endgroup$ Commented Nov 24 at 4:09
  • $\begingroup$ @Alex Youcis: ah. this holds for all idempotent elements in $R[X]$, so forces $X$ to be connected, thanks for this point! $\endgroup$ Commented Nov 24 at 9:28

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