Definition (Chowla subspace).
Let $K \subseteq L$ be a field extension and let $A$ be a $K$-subspace of $L$.
We say that $A$ is a Chowla subspace if for every $a \in A \setminus \{0\}$ one has
$$[K(a):K] > \dim A .$$
In particular $1 \notin A$.
This notion was introduced in M. Aliabadi, Discuss. Math. Gen. Algebra Appl. 45 (2025), 135--157.
For a finite extension $K \subseteq L$, let $\mathcal{C}(L/K)$ be the dimension of the largest Chowla subspace.
Problem.
What are some nontrivial upper and lower bounds for $\mathcal{C}(L/K)$ for finite extensions $K\subseteq L$?
A related concept is that of primitive subspaces.
A $K$-subspace $A\subseteq L$ is primitive if $K(\alpha)=L$ for every $\alpha\in A\setminus\{0\}$.
It is shown in M. Aliabadi & K. Filom, J. Algebra 598 (2022), 85--104 that:
Theorem.
Let $K\subseteq L$ be a finite simple extension. Then the largest possible dimension of a primitive $K$-subspace is
$$\mathcal{P}(L/K)= [L:K] - \max_{\,K\subsetneq E\subsetneq L} [E:K],$$
where the maximum is taken over all proper intermediate fields $E$ (and is $0$ if no such $E$ exist).
Remark.
Every primitive subspace is Chowla, so
$$\mathcal{C}(L/K) \;\geq\; \mathcal{P}(L/K).$$
A natural first step is to investigate the case where $K\subseteq L$ is a simple extension.