Extension elements always exist for field extensions?

Question

I'm trying to understand the concept of field extensions.

If $A\subseteq B$ are fields (i.e. $B$ is an extension of $A$), then we can view $B$ as a vector space over $A$ (i.e. $A$ is the field of scalars).

Is it necessary that there exists $a_1,a_2,\ldots,a_n\in B$ such that $B=A(a_1,a_2,\ldots,a_n)$?

Answer 1

This is true exactly when B is a finite degree extension. For instance, the reals are a field extension of the rationals, but you must adjoin infinitely many numbers to the rationals to get the reals.