Roots of elements for groups over local fields

TL;DR

This work analyzes when the $k$-th power map $P_k$ on linear algebraic groups has dense image or is surjective across real, non-Archimedean, and global fields. It provides a Cartan-subgroup–based criterion for density on real groups and a root-existence framework over non-Archimedean fields showing that all $k$-th roots force unipotent power or containment in a $1$-parameter subgroup; in positive characteristic, such conditions collapse to the identity. The results extend to global fields via local-global methods, yielding parallel conclusions about unipotency and triviality under universal root conditions. Together, they illuminate how Cartan structure, exponential-type dynamics, and global arithmetic govern power-map behavior on algebraic groups.

Abstract

Let $\mathbb F$ be a local field and $G$ be a linear algebraic group defined over $\mathbb F$. For $k\in\mathbb N$, let $g\to g^k$ be the $k$-th power map $P_k$ on $G(\mathbb F)$. The purpose of this article is two-fold. First, we study the power map on real algebraic group. We characterise the density of the images of the power map $P_k$ on $G(\mathbb R)$ in terms of Cartan subgroups. Next we consider the linear algebraic group $G$ over non-Archimedean local field $\mathbb F$ with any characteristic. If the residual characteristic of $\mathbb F$ is $p$, and an element admits $p^k$-th root in $G(\mathbb F)$ for each $k$, then we prove that some power of the element is unipotent. In particular, we prove that an element $g\in G(\mathbb F)$ admits roots of all orders if and only if $g$ is contained in a one-parameter subgroup in $G(\mathbb F)$. Also, we extend these results to all linear algebraic groups over global fields.

Theorems & Definitions (26)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Definition 2.1
• Lemma 2.2
• Lemma 2.3
• proof
• Lemma 2.4
• proof
• Lemma 3.1