Feel free to skip to 10:28 to see how to develop Vladimir Arnold's amazingly beautiful argument for the nonexistence of a general algebraic formula for solving quintic equations! This result, known as the AbelRuffini theorem, is usually proved by Galois theory, which is hard and not very intuitive. But this approach uses little more than some basic properties of complex numbers. (PS: I forgot to mention Abel's original approach, which is a bit grim, and gives very little intuition at all!)

00:00 Introduction
01:58 Complex Number Refresher
04:11 Fundamental Theorem of Algebra (Proof)
10:28 The Symmetry of Solutions to Polynomials
22:47 Why Roots Aren't Enough
28:29 Why Nested Roots Aren't Enough
37:01 Onto The Quintic
41:03 Conclusion

Paper mentioned: https://web.williams.edu/Mathematics/...
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