<p>Geez, OK my “edit” function is giving trouble - I always get readers mad because my posts are in fragments. SORRY!</p>
<p>Anyway, I should address that you wanted also, beyond exciting students about abstract algebra, to get them to realize what it’s like. Well, I’d sort of explain for instance the magic behind figuring out Fermat’s Little Theorem. I’d say that abstract algebra tells you to forget about all the structure of an object EXCEPT what matters for our problem. And lets you solve problems about different objects (like the integers) using their structure. Then, perform your magic and give some examples – POOF, here’s a proof of a cool fact. </p>
<p>The point is, I believe one can show how abstract algebra is a refined theory without delving into messy details, and get a layman to want to take it!</p>