Consider an algebraic extension $K$ of $\mathbb{Q}$.
The degree $[K:\mathbb{Q}]$ of $K$ is defined as the dimension of the extension considered as a vector space.
Now, let $\overline{\mathbb{Q}}$ be algebraic closure of $\mathbb{Q}$.
My question is,
Can we built an arbitrary algebraic extension $F$ of $\mathbb{Q}$ such that $[\overline{\mathbb{Q}}:F]=n$, for any $n\in \mathbb{N}$? How?
