But why can't the quintic be solved? (MEETING AT GOOGLE CHELSEA MARKET)

Our speaker Michael O'Connor, a member of Math for Math's Sake and also the speaker at several past meetings, writes:

You may know that there are formulas to solve quadratic polynomials, as well as polynomials of 3rd and 4th degree, but there is none to solve 5th degree polynomials. This is called Abel's theorem: Abel proved in 1824, Galois proved it in more generality in 1830, and V.I. Arnold gave a simpler, more visual proof in 1963.

I think Arnold's proof is, while not completely simple, simple enough to go through in an hour session, so let's try it! The only thing it requires is knowledge of what complex numbers are.

The high level idea of the proof is: there's no formula in radicals for a fifth degree polynomial because if you move the coefficients of a fifth degree polynomial around in loops, you can get the roots to move around in ways that can't be replicated by a formula in radicals.

To get some intuition for this, I suggest going to: https://duetosymmetry.com/tool/polynomial-roots-toy/ . This website will let you play with moving coefficients of polynomials around and see what happens to the roots.

The best writeup of the proof that I've seen is here: https://web.williams.edu/Mathematics/lg5/ArnoldQuintic.pdf . I originally saw the proof in Chapter 5 of this book, which is excellent in general: http://www.personal.psu.edu/sot2/books/taba.pdf

There's a youtube video which goes through this proof as well. It's only 15 minutes long! Don't worry if it goes too fast, we'll go more slowly during this session: https://youtu.be/RhpVSV6iCko