  Let $\Gamma$ be an Abelian group and let $G$ be a simple graph. We say that
$G$ is $\Gamma$-colorable if for some fixed orientation of $G$ and every edge
labeling $\ell:E(G)\rightarrow \Gamma$, there exists a vertex coloring $c$ by
the elements of $\Gamma$ such that $c(y)-c(x)\neq \ell(e)$, for every edge
$e=xy$ (oriented from $x$ to $y$).
  Langhede and Thomassen proved recently that every planar graph on $n$
vertices has at least $2^{n/9}$ different $\mathbb{Z}_5$-colorings. By using a
different approach based on graph polynomials, we extend this result to
$K_5$-minor-free graphs in the more general setting of field coloring. More
specifically, we prove that every such graph on $n$ vertices is
$\mathbb{F}$-$5$-choosable, whenever $\mathbb{F}$ is an arbitrary field with at
least $5$ elements. Moreover, the number of colorings (for every list
assignment) is at least $5^{n/4}$.
