Given an $n$-mode tensor $\mathcal{T}\in\mathbb{R}^{d_1\times\dotsb\times d_n}$, there exists a Tucker decomposition of $\mathcal T$ of the form $$\mathcal{T} = \mathcal{X}\times_1 W_1\times_2\dotsb\times_n W_n,$$ where $\times_k$ is the mode-$k$ product of a tensor with a matrix.
My question is simple: is the existence of such a decomposition when $\mathcal{T}\in\mathbb{F}^{d_1\times\dotsb\times d_n}$ where $\mathbb{F}$ is an arbitrary field guaranteed/known?
If so, I suppose the argument won't work the same as for higher-order SVD variants (the special case of Tucker when the $W_i$ are orthogonal) for real or complex valued tensors; these are formed by just taking the SVD of each unfolding of $\mathcal{T}$ along its modes. This procedure is specific to real or complex as far as I know.
Any references are appreciated. I have searched quite a bit and come up empty.
