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I know that Galois Theory can be used to answer the following question:

Why is there no formula for the roots of a fifth (or higher) degree polynomial equation in terms of the coefficients of the polynomial, using only the usual algebraic operations (addition, subtraction, multiplication, division) and application of radicals (square roots, cube roots, etc)?

Can real analysis be used also to answer this question (e.g using the intermediate value theorem, etc..)? Is there some sort of connection between real analysis and galois theory?

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There is differential Galois theory, – shamovic May 1 '11 at 16:54

Galois theory is applicable to fields which are neither subfields nor extensions of $\mathbb{R}$ or $\mathbb{C}$, so that's one reason why one should not expect to be able to translate a Galois-theoretic proof into a real/complex-analytic proof.

The main problem with using analytical tools to solve the problem of the quintic is that the tools of analysis are not tuned specifically to handle algebraic functions. Sure, you need complex analysis to prove that every polynomial with complex coefficients has a complex root (assuming you are defining $\mathbb{C}$ analytically), and with a little more effort you could perhaps even prove that the roots, in some sense, vary continuously with the coefficients. But the question is, do they vary in an algebraic way? Here complex analysis falls down, because there isn't (as far as I know) an easy analytical way to distinguish between an algebraic function and a non-algebraic function. (What's the essential analytical difference between, say, $\sqrt{x}$, $x^3$, $\frac{1}{1+x}$ and $\log x$, $\exp x$, or $\tan x$?)

On the other hand, Galois theory enables us to distinguish between, say, $e$ and $\pi$ from $\sqrt[3]{2}$ and $\frac{1+\sqrt{5}}{2}$, and most importantly for the problem of the quintic, between the unique real root of $x^5 + 23 = 0$ (which can be written in terms of radicals) and the unique real root of $x^5 - 15 x + 23 = 0$ (which cannot). The key is the group of automorphisms of the field extension that the roots of these equations lie in, and this is an inherently algebraic concept.

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By "Real Analysis" you are specifically pointing out the properties of Real numbers which are generally uncommon(?) to other fields i.e. the least upper bound property and the greatest lower bound property. But the proof of the Abel Ruffinis theorem doesnt use such a thing. So in that point of view I suppose it has no relation with real analysis. But note that my answer is as vague as the question.

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