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.  Thanks !
Let $G$ be a simply connected algebraic group defined over $C$.   Note : I see the definition of simply connected as "Every isogeny to $G$ is an isomorphism" as given in Hochschild's  "Basic Theory of Algebraic Groups and Lie algebras".   $C$ is an algebraically closed field.   Please help me realise the follwoing things provided they are true:   $G_C$ (the $C$-points of $G$) is simply connected as well. $Lie(G) \cong Lie(G_C)$    Thanks !
Let $R$ be a commutative ring with 1 then why does $a\in N(R) \Rightarrow 1+a\in U(R)$? I am learning Linux and I was trying the gzip command. I tried it on a folder which has a hierarchy like  Personal/Folder1/file1.amr Personal/Folder2/file2.amr Personal/Folder3/file3.amr Personal/Folder4/file4.amr   I ran     "gzip -r Personal" and now its like  Personal/Folder1/file1.amr.gz Personal/Folder2/file2.amr.gz Personal/Folder3/file3.amr.gz Personal/Folder4/file4.amr.gz   How do I go back? Sum of compact sets Let $A,B$ two non-empty compact subsets of a normed space X. How can we prove that the set $S=A+B=\{a+b : a \in A, b \in B\}$ is compact?  Here's my reasoning:  Let $\Omega = \{\Omega_1, \Omega_2,…\}$ be an open cover of $S$. $\Omega$ induces two open cover $X,Y$ respectively of $A,B$, where   $X_i = \{a \in A : a+b \in \Omega_i~~for~some~b\}$   $Y_i = \{b \in B : a+b \in \Omega_i~~for~some~a\}$  in practice $\Omega_i = X_i + Y_i$.  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How to find number of prime numbers between two integers I have two integers, x and y so that x &lt; y. How many prime numbers are there between x and y (exclusive). Is there a formula or algorithm to compute? Determine $\text{Aut}(S)$ where $S = \mathbb{Q}[x]/(x^2)$ Here's the problem: if $S = \mathbb{Q}[x]/(x^2)$, compute the group  of automorphisms $\mbox{Aut}(S)$.   First off, I have the solution and I know it's isomorphic to the multiplicative group $\mathbb{Q}\backslash\{0\}$, so I'm trying to understand it. It uses what is called the substitution principle and the fact that the restriction of any endomorphism $\phi: \mathbb{Q}[x]\to \mathbb{Q}[x]$ to $\mathbb{Q}$ is the identity. It says that these two facts implies that the endomorphisms of $\mathbb{Q}[x]$ are in 1-1 correspondence with $h \in \mathbb{Q}[x]$, where the endomorphism corresponding to $h$ is $\phi_h : \mathbb{Q}[x] \to \mathbb{Q}[x]$ given by $\phi_h(f) = f(h)$ for $f \in \mathbb{Q}[x]$. I guess I just don't understand how it applied the substitution principle? It's probably obvious, but for some reason I am not getting it.  An explanation would be greatly appreciated! Or if you have a different solution, that'd be cool too. Thanks!! Shape with a slanted side (responsive) Can I throw away contents of my Steam Inventory? I have a number of items in my Steam inventory that are never going to get used; for example, some AirMech 2 "gifts" that are marked as NO LONGER VALID.  Is there some way to get rid of them so I don't have them cluttering up my inventory?  Obviously I could do a one-sided trade with a friend, but then they'd have to deal with it instead, and I don't have any alt accounts. How to determine $\int x^2 e^{-x^2}\mathrm dx$? I was wondering how to determine $$\int x^2 e^{-x^2}\mathrm dx?$$ I tried to move the exponential into the integrator, and apply integration by parts, but that will make the integrand more complicated. Thank you! Someone flagged my question as already answered, but it's not I asked a question regarding a very specific feature of a program and whether or not it works with a larger program.  Within hours someone had marked it with "This question may already have an answer here of a similar question, but not related to my question.  The question I asked IS NOT ADDRESSED in the referred-to question. It's frustrating to spend half an hour putting together an intelligent and specific question, and then have someone mark it as a duplicate (or possible duplicate), which apparently causes people to skip over my question. I haven't received any more responses since the question was flagged.   Two questions:  (1) How do I get this flagging removed and (2) what's the best way to prevent these "helpful" flaggings in the future? Is zero input impedance good or possible when considering impedance matching? 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