Why Theres No Quintic Formula (proof without Galois theory) @notallwrong

not all wrong Feel free to skip to 10:28 to see how to develop Vladimir Arnold's amazingly beautiful argument for the non-existence of a general algebraic formula for solving quintic equations! This result, known as the Abel-Ruffini 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/lg5/394/ArnoldQuintic.pdf
Video mentioned: youtu.be/RhpVSV6iCko

updated 3 years ago

Why Theres No Quintic Formula (proof without Galois theory)

not all wrong 2021-07-05 | Feel free to skip to 10:28 to see how to develop Vladimir Arnold's amazingly beautiful argument for the non-existence of a general algebraic formula for solving quintic equations! This result, known as the Abel-Ruffini 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/lg5/394/ArnoldQuintic.pdf
Video mentioned: youtu.be/RhpVSV6iCko

1.5 x 10^80 a day keeps the doctor away | Elliptic Curves

not all wrong 2020-09-01 | How much fruit is too much fruit, and what does that have to do with elliptic curves? The answers are, respectively, 154476802108746166441951315019919837485664325669565431700026634898253202035277999, and a lot.

This is a contribution to the #MegaFavNumbers project, and a first venture into the world of educational videos for its creator, so please let me know what you think. (Sure wish I'd found out about the project more than 3 days before the deadline... Oops.)

The big idea is to look for positive integer solutions to a fairly harmless-looking equation, a/(b+c) + b/(c+a) + c/(a+b) = 4. But it turns out that the simplest such solution has a equal to an 81 digit number: 154,476,...,277,999. That's this channel's MegaFavNumber!

In the video, we explain (using almost nothing more than substituting and rearranging polynomials) how to find this number. It involves an introduction to the exciting world of elliptic curves, also famous in cryptography!

Paper mentioned in the video: http://publikacio.uni-eszterhazy.hu/2858/1/AMI_43_from29to41.pdf