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I've recently been watching some 3blue1brown videos and I watched one about the Monster set in group theory. Now, I have my questions about that but most I'm curious about something else he mentioned, namely quintic equations. I probably spelled that wrong, but what I mean is, an equation one-step above a quadratic equation. For example, something that looks like this: $$ 6x^{5}+3x^{4}+3x^{3}+5x+6=0 $$ So, my question is, what is the universal formula for solving an equation like this. And, where exactly does it come from?

Jean-Claude Arbaut
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HalTheDice360
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  • There is no universal formula for a quintic equation, as shown in Galois theory, which involves group theory. And by the way, quintic is one degree higher than quartic – J. W. Tanner Oct 02 '20 at 19:07
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    @J.W.Tanner No universal formula with only radicals. Otherwise, see https://en.wikipedia.org/wiki/Quintic_function#Beyond_radicals – Jean-Claude Arbaut Oct 02 '20 at 19:10
  • People will tell you that there is no general formula for this - indeed they have at the time of writing this comment. However, the accurate statement is that there is no general formula in radicals to express the roots of a quintic - some new idea must be involved. Of course some quintics do have (radical, rational, integer) solutions, but not all. And even for quartics it is not so much the solution which is useful to know as the method. – Mark Bennet Oct 02 '20 at 19:12
  • Since you are referring to a video, it would be best to ask that at youtube. –  Oct 02 '20 at 19:12
  • @Jean-ClaudeArbaut Spot on, and great link. – Mark Bennet Oct 02 '20 at 19:13
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    @J.W.Tanner, thanks for that correction, I wasn't really thinking about that when I was writing this... – HalTheDice360 Oct 05 '20 at 00:30

2 Answers2

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There is no formula to find the roots of a general quintic polynomial. The Abel-Ruffini Theorem is the formal statement of this: https://en.m.wikipedia.org/wiki/Abel%E2%80%93Ruffini_theorem

Andy Walls
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There is no general formula to solve a quintic, but the proof that no such formula exists requires some machinery. It is one of the famous results of a branch of abstract algebra called Galois theory. See this answer for more details.

marcelgoh
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