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?