Grant Sanderson is a mathematician who is the author of the YouTube channel “3Blue1Brown”, viewed by millions for its beautiful blend of visual animation and mathematical pedagogy. His channel covers a wide range of mathematical topics, which to name a few include calculus, quaternions, epidemic modeling, and artificial neural networks. Grant received his bachelor's degree in mathematics from Stanford University and has worked with a variety of mathematics educators and outlets, including Khan Academy, The Art of Problem Solving, MIT OpenCourseWare, Numberphile, and Quanta Magazine.    
 


In this episode, we discuss the famous unsolvability of quintic polynomials: there exists no formula, consisting only of finitely many arithmetic operations and radicals, for expressing the roots of a general fifth degree polynomial in terms of the polynomial's coefficients. The standard proof that is taught in abstract algebra courses uses the machinery of Galois theory. Instead of following that route, Grant and I proceed in barebones style along (somewhat) historical lines by first solving quadratics, cubics, and quartics. Along the way, we present the insights obtained by Lagrange that motivate a very natural combinatorial question, which contains the germs of modern group theory and Galois theory and whose answer suggests that the quintic is unsolvable (later confirmed through the work of Abel and Galois). We end with some informal discussions about Abel's proof and the topological proof due to Vladimir Arnold.
 


Part I. Introduction


00:00:Introduction


00:52: How did you get interested in math?


06:30: Future of math pedagogy and AI 


12:03: Overview. How Grant got interested in unsolvability of the quintic


15:26: Problem formulation


17:42: History of solving polynomial equations


19:50: Po-Shen Loh 



Part II. Working Up to the Quintic


28:06: Quadratics


34:38 : Cubics


37:20: Viete’s formulas


48:51: Math duels over solving cubics: del Ferro, Fiorre, Tartaglia, Cardano, Ferrari


53:24: Prose poetry of solving cubics


54:30: Cardano’s Formula derivation


1:03:22: Resolvent 


1:04:10: Why exactly 3 roots from Cardano’s formula?



Part III. Thinking More Systematically


1:12:25: Takeaways and Lagrange’s insight into why quintic might be unsolvable


1:17:20: Origins of group theory?


1:23:29: History’s First Whiff of Galois Theory


1:25:24: Fundamental Theorem of Symmetric Polynomials


1:30:18: Solving the quartic from the resolvent


1:40:08: Recap of overall logic



Part IV. Unsolvability of the Quintic


1:52:30: S_5 and A_5 group actions


2:01:18: Lagrange’s approach fails!


2:04:01: Abel’s proof


2:06:16: Arnold’s Topological Proof


2:18:22: Closing Remarks



Further Reading on Arnold's Topological Proof of Unsolvability of the Quintic:


L. Goldmakher. https://web.williams.edu/Mathematics/lg5/394/ArnoldQuintic.pdf


B. Katz. https://www.youtube.com/watch?v=RhpVSV6iCko


 


Twitter: @iamtimnguyen
 


Webpage: http://www.timothynguyen.org
 


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