As title, Is any finite-dimensional extension of a field, say $F$, algebraic and finitely generated?
Say if $K/F$ is a finite extension when $K$ is a finite-dimensional vector space of $F$. Clearly, this implies that $K$ is finitely generated (as an algebra) over $F$, since a basis is a generating set. So every finite extension is finitely generated.
So indeed they all are, is my logic correct?