Image of square map on reductive group

Question

$\DeclareMathOperator\ss{ss}$Let $G$ be a reductive group over a field $F$ of characteristic 0. (Here not necessarily $F=\overline{F}$.) Consider the square map $$ G(F)\longrightarrow G(F), \quad g\mapsto g^2. $$

I know this map is not surjective in general even when $F=\overline{F}$. But how big is the image? Is the image Zariski open?

Also what if one restricts to semisimple elements? Namely, let $G(F)^{\ss}$ be the set of semisimple elements. Is the map $$ G(F)^{\ss}\longrightarrow G(F)^{\ss}, \quad g\mapsto g^2, $$ surjective, say, when $F=\overline{F}$?

Answer 1

The answers to both questions are Yes.

Let $F$ be a field of characteristic 0 and ${\overline F}$ be a fixed algebraic closure of $F$. Let $G$ be a connected reductive $F$-group. By abuse of notation we identify $G$ with the set of ${\overline F}$-points $G({\overline F})$. Let $G_s$ denote the subset of semisimple elements in $G$. For a natural number $n>1$ we consider the map $$\varphi_n\colon G\to G,\quad g\mapsto g^n.$$

Proposition 1. $\varphi_n(G_s)=G_s$.

Proof.
Let $s\in G_s$. It is known that $s\in T$ for some maximal torus $T\subseteq G$. For an element $s\in T$, it is clear that there exists an element $s'\in T\subseteq G_s$ such that $(s')^n=s$, which proves the proposition.

Theorem 2. The set $\varphi_n(G(F))$ is Zariski-dense in $G$.

Proof. Let $U\subseteq G$ be a nonempty open subset. Consider the open subset $U_n:=\varphi_n^{-1}(U)\subseteq G$. We show that $U_n$ is nonempty. Indeed, consider the subset of regular semisimple elements $G_r\subseteq G_s\subseteq G$. It is known that $G_r$ is open and nonempty. Thus $U\cap G_r$ is open and nonempty. Since $U\cap G_r\subseteq G_s$, by Proposition 1 $\varphi_n^{-1}(U\cap G_r)$ is nonempty. Thus $U_n=\varphi_n^{-1}(U)$ is indeed nonempty.

By Corollary 18.3 in Borel's book (2nd edition) the set $G(F)$ is Zariski-dense in $G$. Thus there exists an $F$-point $g\in G(F)\cap U_n$. Then $g^n=\varphi_n(g)\in \varphi_n(G(F))\cap U$. Thus $\varphi_n(G(F))$ is Zariski-dense in $G$, as required.