Suppose I have some complex (unit) vector $V$ on a vector space $\mathcal{H}$ of dimension $N$, where $N$ has prime factorisation $N=\prod^d_i p_i$ (multiplicity is allowed- the $p_i$ need not be unique). $\mathcal{H}$ is spanned by a certain choice of orthnormal basis $\{\mathbb{e}_i\}$. How can one show whether there exists some factorisation $\mathcal{H}=\mathcal{h}_1\otimes\mathcal{h}_2\otimes...\otimes\mathcal{h}_d$, where $h_i$ is a vector space of dimension $p_i$, such that $V$ and all $\{\mathbb{e}_i\}$ are separable?
Alternative perspective: I have a set of $N$ complex numbers $\{z_1, z_2,...,z_N\}$. How does one determine whether this set is fully factorisable- that is, there exist sets of numbers $\{a_1,...a_{p_1}\}, \{b_1,...b_{p_2}\},...,\{x_1,...x_{p_d}\}$ such that each element $z_i$ can be expressed as $z_i=a_{k_1}b_{k_2}...x_{k_d}$ (for a unique set of indices $\{k_1,k_2,...,k_d\}$ for each $z_i$)?