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The algebraic form of Hilbert's 13th Problem asks for the resolvent degree$\text{rd}(n)$ of the general polynomial $f(x) = x^n + a_1 x^{n-1} + \ldots +a_n$ of degree $n$, where $a_1, \ldots, a_n$ are independent variables. Theresolvent degree is the minimal integer $d$ such that every root of $f(x)$ canbe obtained in a finite number of steps, starting with $\mathbb C(a_1, \ldots,a_n)$ and adjoining algebraic functions in $\leq d$ variables at each step.Recently Farb and Wolfson defined the resolvent degree $\text{rd}_k(G)$ of anyfinite group $G$ and any base field $k$ of characteristic $0$. In this setting$\text{rd}(n) = \text{rd}_{\mathbb C}(S_n)$, where $S_n$ denotes the symmetricgroup. In this paper we define $\text{rd}_k(G)$ for every algebraic group $G$over an arbitrary field $k$, investigate the dependency of this quantity on $k$and show that $\text{rd}_k(G) \leq 5$ for any field $k$ and any connected group$G$. The question of whether $\text{rd}_k(G)$ can be bigger than $1$ for anyfield $k$ and any algebraic group $G$ over $k$ (not necessarily connected)remains open.

Hilbert's 13th Problem for Algebraic Groups