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.
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asked Aug 26, 2021 at 11:55
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1$\begingroup$ Look here $\endgroup$– PeterAug 26, 2021 at 12:13
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1$\begingroup$ Solving by radicals might be helpful when working in other fields. For example, the roots of $x^2-x-1$ are $\frac{1\pm\sqrt 5}{2}$ not only in $\Bbb R$, but also in finite fields $\Bbb F_p$, as long as $\sqrt 5$ make sense (e.g., we can readily find the roots of $x^2-x-1$ in $F_{31}$ because $6^2=36\equiv 5$) $\endgroup$– Hagen von EitzenAug 26, 2021 at 12:15
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1$\begingroup$ @HagenvonEitzen But is that about radicals specifically? $\sqrt{5}$ is just a solution to $x^2-5$. So doesn't this carry across to solutions of arbitrary polynomials, even if they're not solvable by radicals? $\endgroup$– Thomas AntonAug 26, 2021 at 12:18
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1$\begingroup$ You suggest that solvability by radicals is unmotivated. I disagree. When we seek solutions to polynomial equations, it is natural to wish for the solutions to be as simple as possible. It is therefore natural to hope for solutions to be obtained using 'elementary' operations as addition, multiplication and exponentiation. $\endgroup$– fwdAug 26, 2021 at 12:44
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1$\begingroup$ @fwd Why not throw in logarithms? Where do operations stop being elementary? And it's certainly natural to hope, but once you realise that hope is unfounded, then why keep talking about solvability by radicals? $\endgroup$– Thomas AntonAug 26, 2021 at 12:53
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1 Answer
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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.
