This paper introduces a copula based multivariate rank test for independence extending existing approaches from literature to p dimensions. Then, a multiparametric p-dimensional generalization of the FGM copula is provided that can model the behavior in each vertex of the p-dimensional unit cube using exactly one parameter per vertex - the family of polynomial copulas. The independence copula is nested in this family if and only if every parameter is zero. In this case, a popular way to test for independence is comparing an estimate of the vector of parameters to a vector containing zeros only. Unfortunately, due to the mere quantity of parameters, no established estimation procedure can be used in higher dimensions. Instead, the developed multivariate rank test is applied sequentially to every parameter to test for joint squared deviation from independence. Applying this new test to the polynomial copula results in the new vertex test which is a test for independence with focus on the high dimensional tail regions. It is compared to similar nonparametric rank tests of independence by means of calculation time and power under several alternatives and sample sizes.