Complexity of Tensor Rank over an Infinite Field
There is a recent preprint about this: http://galton.uchicago.edu/~lekheng/work/np.pdf . It shows that most rank-related issues with tensors are NP hard over $\mathbb{R}$ and $\mathbb{C}$. (It also mentions that deciding the rank over $\mathbb{Q}$ is NP hard.)