Is there a field extension $E$ containing $S$ which has finite dimension when considered as a vector space over $S?$

Question

An algebraic number field (or simply number field) is an extension field $K$ of the field of rational numbers $\mathbb {Q}$ such that the field extension $K/\Bbb Q$ has finite degree (and hence is an algebraic field extension). Thus $K$ is a field that contains $\mathbb {Q}$ and has finite dimension when considered as a vector space over $\mathbb {Q}.$

Consider:

$$ \exp: \Bbb Q \to (S \subset \Bbb R^+) $$

$(S,\times)$ is a group. Furthermore $(S,\times, *)$ is a field by:

$$ m: S \times S \to S $$ with $(e^a,e^b)$ to $e^{ab},$ where $a,b \in \Bbb Q.$

Is there a field extension $E$ containing $S$ which has finite dimension when considered as a vector space over $S?$

Answer 1

Yes, such an extension exists. One possibility is to take any extension field $K$ of $\mathbb{Q}$, given you have one at hand: $K$ is, up to isomorphism, a field extension of $S$, because $S$ is, up to isomorphism, just $\mathbb{Q}$.

An easy possibility, if I understand your question correctly, would be $E = S$. This is trivially a field extension of $S$ of dimension $1$. Please give me a hint if I didn't understand your question correctly.