example for algebraic but not finite dimensional vector space 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
 A: An extension $K/F$ is said to be algebraic if all of the elements of $K$ are algebraic over $F$, i.e. the solutions of polynomials in $F[X]$. Let's take the field extension $\mathbf{Q}(\sqrt{2},\sqrt{3},\ldots, \sqrt{n},\ldots)$. Every element we adjoin is algebraic. Indeed, $\sqrt{n}$ satisfies $x^2-n=0$. So, this extension is algebraic over $\mathbf{Q}$, but it certainly isn't finite. 
A: First example: Let $K:=\mathbb{Q}$ be the field of rationals;
and let $K:=\bar{\mathbb{Q}}$ be it's algebraic clousure.
Then $F/K$ is algebraic but it is not finite dimentional.


Second example: Let $K:=\mathbb{Q}$ be the field of rationals;
and let's consider an infinite set of algebraic elements $A$; such that any finite set of $A$ is algebraicly independent. Then let $K:=\mathbb{Q}(A)$.
If you let 
$A= \bigg{\{} \sqrt{p} \ : \ p \ \text{is a prime number} \ \bigg{\}}$;
and let $K:=\mathbb{Q}(A)$; then you will get exactly the answer of Antonios-Alexandros Robotis.
Note that $\mathbb{Q}(A)=\mathbb{Q}(A')$; where 
$A'= \bigg{\{} \sqrt{n} \ : \ n \in \mathbb{N} \ \bigg{\}}$.
Also if you let
$A=$ set of all algebraic elements;
then you will get the first example.

Now you can construct another examples;  


*

*$A= \bigg{\{} \sqrt[3]{p} \ : \ p \ \text{is a prime number} \ \bigg{\}}$; 

*$A= \bigg{\{} \sqrt[2p+1]{p} \ : \ p \ \text{is a prime number} \ \bigg{\}}$. 
A: $\Bbb F$ 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.
