A threeld is a generalization of a field, with three operations, such that the $F$ is a field with respect to the first (outer) and second (middle) operations (call it the outer field), and $F\setminus\{0\}$ is a field with respect to the second and third (inner) operations (the inner field). The following definitions and results are originally from https://cp4space.hatsya.com/2022/05/25/threelds/
If the outer field does not have characteristic $2$, then the element $-1$ is the only element of order two with respect to the middle operation, and therefore with the exception of the field $\mathbb{F}_3$ this cannot form a threeld (the characteristics would not agree). Therefore, all other threelds have outer characteristic $2$. For finite threelds, this thus restricts threelds to $\mathbb{F}_{2^p}$, where $2^p-1$ is a Mersenne prime.
Infinite threelds of arbitrary cardinality exist (see the article for proof via the Löwenheim–Skolem theorem). Countable examples must have the middle operation isomorphic to a vector space over $\mathbb{Q}$. My question is, is it possible for that vector space to be finite dimensional? If so, is there an explicit construction of this?
