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Following is what I understand regarding the simply connected linear algebraic groups afer reading some definition in Hochschild's 'Basic Theory of Algebraic Groups' :
(I don't know about fundamental group)
Let $G$ be a linear algebraic group over a field $F$ of characterstic zero. Then $G$ is said to be simply connected if any surjective homomorphism of algebraic groups $$\phi : H \to G$$ where Ker $\phi$ is finite turns out to be an isomorphism.
Now, $G$ being defined over $F$ means the defining equations have coefficients in $F$. However, $G$ has points coming from $F^n$ as well as $K^n$ where $K$ is some extension of $F$. Let us denote the $F$-points in $G$ by $G_F$.
Following are my questions:
What can be say about the simple-connectness of $G_F$? Is this simply connected as well ? I would appreciate if you could refer to some results leading to the conclusion.
What about the Lie algebra of $G_F$? Is $Lie(G) \cong Lie(G_F)$?
I do not know much algebraic geometry and these type of results are coming in a paper which I am reading. Please help me out. I would appreciate if the explanation is simple and detailed.