How to show a class of structures is not axiomatizable? For example, let $F$ be a field and $L$ be the language of $F$-vector space.
(1) Prove that the class of finite dimensional $F$-vector space is not axiomatizable.
(2) Prove that if $F$ is infinite then the class of infinite dimensional $F$-vector space is not axiomatizable.
Or let $L$ be the language of rings. 
(3) Prove that the class of algebraic extensions of $\mathbb{Q}$ is not axiomatizable. 
I think the common way to prove this type of statement would be: first suppose the class is axiomatizable. Then there is some $L$-theory axiomatizing the class. We make a new language $L'$ by adding new symbols to $L$ then construct a $L'$-theory $T'$ and show $T'$ is consistent by compactness then observe a contradiction.
However, this method requires a bit of algebra knowledge. Can anyone give some hints for the above problems I have listed?

Hint :  for all these questions, you can use the ascending Löwenheim-Skolem theorem. 
