
> 
> Is there a division algebra $D$ with center $K$ that satisfies the
>  following 3 conditions?
> 
> 
> 


1) $D$ is of infinite dimension over $K$;


2) every element of $D$ is algebraic over $K$;


3) $D$ is finitely generated (as division $K$-algebra).



Infinite dimensional finitely generated algebraic division algebra
This is a fairly well known old and open (as far as I know) problem: Kurosh’s Problem for division rings. See, for example, Question 3 in Agata Smoktunovicz’s [2006 ICM talk](http://icm2006.org/proceedings/Vol_II/contents/ICM_Vol_2_12.pdf).


