Inspired by: Joy in Discovery, Class Project

Your task: Prove either of the following statements:

• There exists a polynomial $p\left(x\right)$ of degree five over the rational numbers, with a root $x$, so that $x$ cannot be obtained from the rationals by field operations and iterated extraction of roots.
• There is no set formulas, given in terms of field operations and iterated extraction of roots, expressing the roots of a fifth-degree polynomial over the rationals in terms of the coefficients.

Note that the former clearly implies the latter, but you are free to prove either.

By "field operations", i mean just $+,-,\times ,/$. By "extraction of roots", i mean operations of the form "^n\sqrt{(-)}$. You are allowed to take even roots of negative numbers, i.e to work inside the complex numbers.

To clarify:

• In the case of the first problem, I am looking for a specific equation of the form $a{x}^5+b{x}^4+c{x}^3+d{x}^2+ex+f$, with $a,b,c,d,e,f$ rational numbers, ($a$ nonzero) which has at least one solution $x$, so that $x$ cannot be written in terms of rational numbers, addition, subtraction, multiplication, division, and taking roots.
• When taking roots, you can also take a non-principal root. For instance, you can take both ${}^5\sqrt{10}$ (the positive real fifth root of ten), but also the four other roots, ${}^5\sqrt{10}\cdot e^{i\frac{2\pi k}{5}}$, for $k=1,2,3,4$.
• The solution $x$ may be a complex number.
• In the case of the second problem, we just want to prove that there is no single formula which works for all fifth-degree polynomials, expressing the roots in terms of the coefficients.

Reading further will give you more information, which may not be in the spirit of things. I will take care not to give too many spoilers, but read carefully.

(You have already been spoiled that the solution involves something called "Galois theory", but if you know what that is, you almost certainly already know it applies to this question).

I just started taking a course on Galois theory for my master's degree in mathematics. The course is pretty far below my level; I am taking it for annoying bureaucratic reasons. Despite that fact, although I have been exposed to the some of the ideas before(which probably made it significantly easier for me), I've never really scratched the surface of Galois theory. But during the first lecture, as the professor started sketching out the direction of the course, how we would [do Galois theory] and use this to show the impossibility of solving the general quintic, it occurred to me that I could pretty much see how the argument would go. After the lecture, I wrote it down. And then I thought back to the posts above, one of which I had recently read for the first time, and I thought: this may actually be an opportunity for someone else to practice.

(It was an opportunity for me to practice, but not as much as it would be for someone who hadn't just been told what input was necessary, as I had been)

From here, anything I consider a spoiler goes in rot13

Requirements/Hints

In the most general sense, this question is solvable with nothing more than very simple 19th-century mathematics - we know, because it was solved by those methods. Nevertheless, if you want to solve this without being as good as Galois and Abel, perhaps some sort of checklist is nice, so you can know if you have a "fair shot" at this. I think anyone with an undergraduate degree in math probably has the necessary basics, but I am not certain about that. I will give the fields of math necessary, in very broad terms, but be warned that one of them may be non-obvious, and so constitute a mild spoiler: "Onfvp nofgenpg nytroen, v.r tebhc gurbel, evat gurbel, svryqf, cbylabzvny evatf.".

More specifically, there are some background concepts that may be hard to invent de novo. But again, these may contain more information than desired, in that it's not obvious that they apply to the problem at all. Nevertheless, they are "Gur abgvba bs "fbyinoyr tebhc". Gur vqrn bs nffbpvngvat n Tnybvf tebhc gb n svryq rkgrafvba, naq gur Tnybvf pbeerfcbaqrapr.."

I am very uncertain about the difficulty of this problem. It depends a lot on your mathematical background. But I don't think it's easy.

Update: After writing out my solution in more details, thinking through some of the things one would have to invent to tackle this, I am now fairly certain that this is a very hard problem.