I've been wanting to ask this for a while now and hope that it will enlighten me on why precisely abstract algebra is powerful.
What were some major unsolved problems in mathematics which were first solved by abstract algebra? Did algebraists anticipate the discoveries (was solving these problems their goal) or were they more later applications of a certain discovery in the field.
I would like to add that to illustrate why the subject of abstract algebra in particular, preferable they were not solved before the invention of abstract algebra. Also that they actually lead to breakthroughs in fields which concern the pure mathematicians, are not just interesting trinkets (it is perhaps slightly subjective). By that I mean interesting one off riddles or rogue applications that don't usually concern the mathematician, as well as interesting mathematical things but that are not more than just interesting.
This is a big-list question, so the more examples the better (unless there happen to already be a lot from other users). The reason is that examples put philosophies into context, allow you to see patterns for yourself, and ultimately having just a few examples is seldom persuasion for entire areas of study.
Edit: The only major problem first solved by abstract algebra I can currently recall is the unsolvability of certain polynomials by radicals.