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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.

Thomas Anton
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  • The Greeks didnt think all numbers were rational! – TonyK Aug 26 '21 at 11:59
  • Eventually, they didn't, I will clarify. – Thomas Anton Aug 26 '21 at 12:00
  • The only merit is that we do not need to estimate the roots (if they happen to be very large) in the case we have a formula. But even in degree $3$, usually numerical methods are applied. This theorem is rather of theoretical interest. – Peter Aug 26 '21 at 12:00
  • Whether a group is solvable, is however of big interest for group theory. The hall theorem is an important theorem depending on the solvability of a group. – Peter Aug 26 '21 at 12:04
  • Could you provide a reference? I am unfamiliar and can only find Hall's Marriage Theorem and a variety of different results about Hall subgroups, none called The Hall Theorem. – Thomas Anton Aug 26 '21 at 12:09
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    Look here – Peter Aug 26 '21 at 12:13
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    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 $6^2=36\equiv 5$) – Hagen von Eitzen Aug 26 '21 at 12:15
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    @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? – Thomas Anton Aug 26 '21 at 12:18
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    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. – fwd Aug 26 '21 at 12:44
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    @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? – Thomas Anton Aug 26 '21 at 12:53
  • For the sake of exactness. And just because there exist polynomial equations not solvable by radicals does not mean we should forget about solving polynomials where possible in terms of the most basic operations. – fwd Aug 26 '21 at 13:09
  • To me the notion is one of several that can be used to classify the "degree of irrationality" of a number, in the same way that Hausdorff dimension classify "degrees of being measure zero", order of growth classify "degrees of growth of functions", smoothness notions classify "degrees of continuity", and many other similar examples one encounters. We have rational numbers, constructible numbers, real-radical numbers, cubic/conic constructible numbers (and analogous higher degree notions), solvable numbers (i.e. expressible using radicals), elementary numbers, Turing computable numbers, etc. – Dave L. Renfro Aug 26 '21 at 13:18
  • There seems to be a wrong underlying belief here that all numbers are real numbers. (Or when pointed out, maybe you would say complex numbers.) Hagen von Eitzen already pointed out finite fields, and I would throw in $p$-adic numbers. – Torsten Schoeneberg Aug 26 '21 at 14:30
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    Also, do I understand correctly that you think Galois theory is motivated, but asking whether a group is solvable is not? Also, as far as I know, historically, Galois theory was discovered precisely because of the problem of solving polynomials with radicals. In hindsight, ruler and straightedge construction fit into this, kind of like in hindsight, abelian groups are the basic case of solvable groups ... – Torsten Schoeneberg Aug 26 '21 at 14:36
  • @TorstenSchoeneberg In my opinion, Galois theory and field theory more generally are motivated because they state many beautiful, deep and surprising things about polynomials, which connect to other areas of mathematics, and polynomials, although they have a strange definition are motivated because they are central to ring theory and analysis. – Thomas Anton Aug 26 '21 at 14:56
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    IIRC, there is a particular function that was invented just to be able to write a closed-form solution for a general quintic. Evaluating the function requires you to solve a quintic, but it's a particular limited type of quintic so it's not completely a circular method. I doubt that you get much traction by throwing the usual closed-form functions such as logarithms and trig functions into the mix. – David K Aug 26 '21 at 17:07
  • For completeness, I think what @DavidK alludes to is the Bring radical, see https://en.wikipedia.org/wiki/Bring_radical. Note that another somewhat equivalent method (discussed in there) are certain elliptic functions. However, before one gets excited as in https://math.stackexchange.com/q/760608/96384, one should look at https://mathoverflow.net/q/61409/27465. I guess there's a hypergeometric series for everything, but just saying "logs or trig functions" is not that helpful either, I agree. As the Overflow post and its answers show, this is tricky. – Torsten Schoeneberg Aug 26 '21 at 22:26
  • @ThomasAnton, out of curiosity, what would be three of those "many beautiful, deep and surprising things about polynomials" which Galois theory teaches us and which have nothing or very little to do with solvability by radicals / solvability properties of the Galois group? – Torsten Schoeneberg Aug 26 '21 at 22:28
  • The Fundamental Theorem of Galois Theory, the most purely algebraic way I know of proving The Fundamental Theorem of Algebra, The Normal Basis Theorem – Thomas Anton Aug 27 '21 at 02:23

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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.

D_S
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