Is there a subfield $K$ such that $\mathbb{Q} \subset K \subset \mathbb{R}$ (proper subset) as follows:
$\mathbb{R}$ is a vector space over $K$ and has no finite generating set and $K$ is a vector space over $\mathbb{Q}$ and has no finite generating set.
Sum and multiplication are both defined as usual in $\mathbb{R}$.
My guess is that there is one: $K$ should not contain continuum as it is field so in this case it would be equal to $\mathbb{R}$, so $K$ is countable, and $\mathbb{R}$ has no finite generating set over countable sets. On the other hand, $K$ can contain countable but infinite number of elements from $\mathbb{R} - \mathbb{Q}$ such that $\forall a \in \mathbb{R} - \mathbb{Q} \,\,\,\,a \in K \Rightarrow \exists b \in K, q \in \mathbb{Q}:a\neq q\cdot b$, i.e. contains infinite number of irrational numbers so that pairwise their division is not rational. In this case number of elements in $K$ basis is countable, but infinite (each irrational number is in basis, if its product with some rational number isn't in it). $K$ seems to be field and not equal to $\mathbb{R}$, but I'm not sure.
Thanks in advance!