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.

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    $\begingroup$ Are you familiar with Galois theory and (un)solvability of polynomial equations in radicals? $\endgroup$ Jun 23 '20 at 22:15
  • $\begingroup$ @MoisheKohan thanks I've edited it. I can't believe I forgot that - I had meant to add it when I started typing the question. I do not believe I know of any others unless my memory fails me. $\endgroup$ Jun 23 '20 at 22:20
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    $\begingroup$ On a related note, the impossibility (using straightedge and compass) of doubling a cube or trisecting an angle, as well as the determination of which regular polygons are constructible, were all proved using abstract algebra. See Pierre Wantzel for more details. $\endgroup$
    – user169852
    Jun 23 '20 at 22:28
  • $\begingroup$ Algebraic number theory is full of such examples. $\endgroup$ Jun 23 '20 at 22:30
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    $\begingroup$ Actually "abstract algebra" as such didn't exist when those problems were solved. Galois studied permutation groups, not abstract groups. The general definition of a group didn't come until 1854 (Cayley). The term "abstract algebra" dates from 1860, but in its modern sense it is from the early 20'th century. $\endgroup$ Jun 23 '20 at 23:19

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