Galois groups of multivariate Tutte polynomials

The multivariate Tutte polynomial ${\widehat{Z}}_M$ of a matroid $M$ is a generalization of the standard two-variable version, obtained by assigning a separate variable $v_e$ to each element $e$ of the ground set $E$. It encodes the full structure of $M$. Let $\text{bv}=\{v_e{\}}_{e\in E}$, let $K$ be an arbitrary field, and suppose $M$ is connected. We show that ${\widehat{Z}}_M$ is irreducible over $K(\text{bv})$, and give three self-contained proofs that the Galois group of ${\widehat{Z}}_M$ over $K(\text{bv})$ is the symmetric group of degree $n$, where $n$ is the rank of $M$. An immediate consequence of this result is that the Galois group of the multivariate Tutte polynomial of any matroid is a direct product of symmetric groups. Finally, we conjecture a similar result for the standard Tutte polynomial of a connected matroid.