Let $R = GF(q), q = p^r$, be a field with identity $e$, where $p$ is a prime number. Let $S=GF(q^n)$ be an extension of $R, n\geq 2$ and $K = GF(q^{mn})$ be an extension of $S$, where $m$ is prime. Let $_RW$ be a subspace of $_RK$ such that $$ \operatorname{dim}_RW =n, e\in W, W\neq S $$ I want to prove that there exist a primitive element of $K$ (that is the element of multiplicative order $q^{mn}-1$) in $W$.

But so far my attempts have not been successful. Experiments on the computer confirm this hypothesis.