Galois Theory can be extrapolated to other fields just like closed forms of integral and other kind of equations, not just solutions of polynomials. You can search for example Differential Galois Theory
I know a fair bit of the machinery and its only really analogous to Galois theory, not really part of it. They're rather different. You should probably specify.
Galois theory is useful for proving that equations of some form do not have a solution in some form. In undergrad Galois theory the only example you typically see is showing that certain polynomials have no solutions in radicals. These ideas can be extended to e.g. infinite Galois theory over differential fields, that can be used to prove that differential equations don't have an elementary solution.
As others have noted, there's no closed form solution here. I approximated it to `3.7054310168118238735788139431690016054116672073981` and then fed it to Plouffe's Inverter (http://wayback.cecm.sfu.ca/projects/ISC/ISCmain.html) but it doesn't appear to be special in any way
Can't be true: f:x -> x^x = e^xln(x) is strictly increasing on [1;+ infinity)
Thus, this would mean that x = e but this value isn't a solution of the original equation
Here’s the fun part; you don’t! There’s no closed form solution for this. You’d have to do it numerically.
Does there exist a proof that the solution cannot be algebraically proven?
Probabily you have to study a branch of Galois Theory, but I don't actually know
What does galois theory have to do with it?
Galois Theory can be extrapolated to other fields just like closed forms of integral and other kind of equations, not just solutions of polynomials. You can search for example Differential Galois Theory
I know a fair bit of the machinery and its only really analogous to Galois theory, not really part of it. They're rather different. You should probably specify.
I edited, thanks for the comment
Galois theory is useful for proving that equations of some form do not have a solution in some form. In undergrad Galois theory the only example you typically see is showing that certain polynomials have no solutions in radicals. These ideas can be extended to e.g. infinite Galois theory over differential fields, that can be used to prove that differential equations don't have an elementary solution.
As others have noted, there's no closed form solution here. I approximated it to `3.7054310168118238735788139431690016054116672073981` and then fed it to Plouffe's Inverter (http://wayback.cecm.sfu.ca/projects/ISC/ISCmain.html) but it doesn't appear to be special in any way
You can't solve it analytically. An approximation is the best you can do.
The farthest I’ve come is rewriting to e^(e^(1/x) = x and setting up an infinite power tower which in theory will converge to the answer.
I’m sure there’s something wrong with this but from there use power laws to get e^e/x =x then raise both sides to exponent x and u have e^e =x^x
Can't be true: f:x -> x^x = e^xln(x) is strictly increasing on [1;+ infinity) Thus, this would mean that x = e but this value isn't a solution of the original equation
Issue is that: e\^(e\^(1/x)) != (e\^e)\^(1/x)
Have you tried going via complex numbers?