Why Should I Care about Solvability by Radicals?

Question

The Egyptians liked studying numbers in terms of Egyptian fractions. The Greeks liked straightedge and compass constructions. Both really liked rational numbers, especially the Pythagoreans before the proof of the existence of irrational numbers. Rational numbers are obviously very important, but knowledge of the existence of irrationals means they're held in slightly lower esteem today. Egyptian fractions became a mere novelty, and later straightedge and compass constructions met the same fate, although the focus in Europe and the Middle-East on Greek classics kept them alive much longer. They're just not particularly natural concepts. To me, solvability by radicals seems to belong with them. Before the Abel-Ruffini Theorem, it could have been a much more important concept, just like before the proof that $\sqrt{2}$ is irrational, rational numbers could have been somewhat more important. But without a statement like the negation of the Abel-Ruffini Theorem, solvability by radicals seems completely unmotivated. Studying it has certainly given us many important, natural constructs, but that doesn't make it important by itself; part of being natural is showing up in many places, so if we hadn't studied solvability by radicals, we'd have come across Galois Theory some other way, and indeed we did, with ruler and straight-edge constructions. So I'd like to know reasons that people would be interested in solvability by radicals, or indeed solvability of groups, for their own sake.

Answer 1

The "simplest" operations are addition, multiplication, taking powers of numbers and their "inverses" (e.g. subtraction, division, and taking roots). It's always been nice to express roots in terms of these intuitive processes, and they are moreover all readily computable e.g. roots by power series.