TITLE: example for algebraic but not finite dimensional vector space
QUESTION [6 upvotes]: there is a theorem on Galois theory that
if F is a finite dimensional extension field of K (say [F:K]=n) then F is algebraic over K
I'm thinking about the converse of this statement. is the converse false? can anyone give me an example that contradict the converse

REPLY [3 votes]: $\Bbb F$ is an algebraic extension of a field $\Bbb K$ if every element $\phi \in \Bbb F$ satisfies some polynomial $k(x) \in \Bbb K[x]$; $k(\phi) = 0$. As such, we need not have $[\Bbb F: \Bbb K] < \infty$, since different elements of $\Bbb F$ may satisfy polynomials over $\Bbb K$ of different degrees.  The most ready example is probably the field of algebraic numbers over $\Bbb Q$, which contains elements satisfying polynomials $p(x) \in \Bbb Q[x]$ of every degree.  See this wikipedia entry for more.