We studied in class that regular languages closed under intersection.
My question is : if we take the irregular language $L =$ {$a^nb^n : n\geq 0$} and the regular finite language $L' = \{a^3 b^3\}$ then $L \cap L'= L'$ and because $L'$ is a regular language so $L$ is also regular by the intersection sentence. I know this is not right, but I don't know what I have missed.
That regular languages are closed under intersection means that if $L,M$ are both regular, then so is $L \cap M$.
You assumed that if $L$ and $L \cap M$ are regular, so is $M$, and that is blatantly false, which your counter-example illustrates.