I have a soft question that will probably get boo-ed.
It struck me recently that I have known since my pre-calculus days that there is no 'general solution' for polynomials with $\mathrm{deg}(p) \geq 5$, but I have never seen the proof. In my grad school days I became aware that this result comes from Galois Theory, but I have never learned any Galois Theory, and don't have any intuition for how the theory would produce such a result. I am always fascinated by results that are elementary enough to be explained even to beginners, but have non-trivial proofs.
My question is, how deep into Galois Theory would one need to go to be able to work through this result? My background is in algebra (first few chapters of Hartshorne, Atiyah & MacDonald, etc.)
Also, the wording used on Wikipedia is that the result implies there is no 'solution by radicals'. Can I take this to mean no closed form solution? Or is the result slightly weaker, e.g. would it make any sense for a polynomial to have a closed form solution to $p(x) = 0$ that is not by radicals?
Not very deep. In modern approach, this is proved in three steps.
First, you prove the fundamental theorem of Galois theory, which is a culmination of somewhat longish path made of very simple and straightforward steps. As the name suggests, this theorem is fundamental to Galois theory, so necessarily cannot be very deep.
Then, you prove that solutions by radicals correspond to decompositions of Galois group with cyclic factors. This is straighforward.
Finally, you show that there exists a polynomial with Galois group of $A_5$, which cannot have decomposition into cyclic factors, as it's already simple. This one is also quick.
Overall, if you have background of Hartshorne and Atiyah-MacDonald, it shouldn't take you more than a few hours to get to the meat of it. I recommend Aluffi's "Algebra: Chapter 0".