Why are many equations not solvable by applying only Elementary functions?

Question

Why are many equations not solvable by applying only Elementary functions?

Elementary functions:

"In mathematics, an elementary function is a function of one variable which is the composition of a finite number of arithmetic operations (+ – × ÷), exponentials, logarithms, constants, and solutions of algebraic equations (a generalization of nth roots). The elementary functions include the trigonometric and hyperbolic functions and their inverses, as they are expressible with complex exponentials and logarithms." (Wikipedia - Elementary function)

Let us discuss this on the example equation x + e^x = 0, x real.

I already know, the reason is that x and e^x are algebraically independent. (Is there a reputable reference for that?)

But what is the the exact and complete reason of the non-solvability of this equation or of such equations only by Elementary functions?

Answer 1

There are just not enough elementary functions to cover even many common equations.

But this problem is more theoretically, because it is easy to get solutions numerically with arbitary precision in most cases.

A similar consequence is that many easy-looking functions do not have a closed-form-antiderivate. Ironically, the function $e^{-x^2}$ , playing a very important role in probability theory, is such a function.

Even polynomials with degree $5$ or higher cannot be generally solved by radicals.

An equation must be very special to have a closed-form-solution.

Answer 2

Let $T_1(z)$, $T_2(z)$ and $T(z)$ denote mathematical terms which contain a variable $z$. Each ordinary equation $T_1(z)=T_2(z)$ of one unknown $z$ can be transformed into an equation $T(z)=c$, with $c$ a constant. If the term $T(z)$ contains only elementary functions, $T$ can be treated as an elementary function. If you want to solve the equation algebraically, you have to invert $T$.

The theorem of Joseph Fels Ritt in Ritt, J. F.: Elementary functions and their inverses. Trans. Amer. Math. Soc. 27 (1925) (1) 68-90 answers which kinds of elementary functions can have an inverse which is an elementary function.