Let $T_1, \ldots, T_n \in GL(n,\mathbb{F}_p)$. Suppose for all $\vec{v} \in \mathbb{F}_p^n$ we have $\det (T_1 \vec{v}, T_2 \vec{v}, \ldots, T_n \vec{v}) = 0$. Now, let $k$ be a finite extension of $\mathbb{F}_p$. Is it true that $\det(T_1 \vec{v}, \ldots, T_n \vec{v})=0$ for all $\vec{v} \in k^n$?

I know the above is true in some special cases. For example, if the $T_i$ are powers of a single linear transformation $T$, if the $T_i$ are the elements of a finite abelian subgroup of $GL(n,\mathbb{F}_p)$ of order equal to $n$ coprime to $p$, or if $\mathbb{F}_p$ is replaced with $\mathbb{Q}$ and $k$ is real.

UPDATE: So the above, unfortunately, isn't true in general. But ultimately, as the comments below indicate, I want want to know about something rather specific (to which the above problem is related): Let $p$ be an odd prime. If $E$ is the group of units of a real abelian number field with Galois group $G$, does the cyclicity of $E\otimes k$ as a $k[G]$-module imply the cyclicity of $E \otimes \mathbb{F}_p$ as an $\mathbb{F}_p[G]$-module. I know that this is true if $p$ does not divide $|G|$.