Under the Axiom of Choice, we can pick a Hamel basis $H$ for $\mathbb{R}$ as a vector space over $\mathbb{Q}$. Adjoining all but some elements of $H$ to $\mathbb{Q}$ shows that for any cardinality $1\leq\kappa\leq c$ there is an intermediate field $\mathbb{R}-F-\mathbb{Q}$ such that the extension $\mathbb{R}:F$ has degree $\kappa$.
Can some part of this result (especially when $\kappa <c$) be obtained without AC?