Nash Equilibria with Derangement Degree Probabilities

We prove for every $n\ge4$ the existence of an $n$-player game in normal form with integer payoffs that has a unique Nash equilibrium, which is fully mixed. In the equilibrium, each probability weight is an algebraic number of degree $\mathbin{!n}$ (the derangement number), and its minimal polynomial has Galois group $S_{\mathbin{!n}}$ and $\mathbin{!n}+1$ nonzero coefficients.