  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. The
resolvent degree is the minimal integer $d$ such that every root of $f(x)$ can
be 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 any
finite 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 symmetric
group. 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 any
field $k$ and any algebraic group $G$ over $k$ (not necessarily connected)
remains open.
