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> For each real $A>0$, let $x\_A$ denote the positive root $x$ of the polynomial $x^5-3x-A$. Is the function $(0,\infty)\ni A\mapsto x\_A$ elementary?
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~~[I am using this [definition of elementary functions](https://en.wikipedia.org/wiki/Elementary_function#Basic_examples), except for the algebraic functions mentioned there.]~~


*Edit:* After the edifying answer by [Timothy Chow](https://mathoverflow.net/a/442656/36721), I would like to clarify the use of the term *elementary functions* in this context just as Timothy Chow suggested: "the functions that can be expressed using some finite combination of constant functions, the identity function, $\exp$, $\log$, composition, and arithmetic operations".



Can the positive root of this polynomial be expressed elementarily?
I believe that the class of functions that Iosif Pinelis is interested in is what I would call *exponential-logarithmic functions* or *EL functions*; that is, they are the functions that can be expressed using some finite combination of constant functions, the identity function, $\exp$, $\log$, composition, and arithmetic operations (${+}{-}{\times}{\div}$). Some authors call this class of functions *elementary functions*, but that term is now more commonly used in a different sense, which includes algebraic functions. But Iosif Pinelis explicitly says he does not want to include all algebraic functions.


[Alexandre Eremenko's answer](https://mathoverflow.net/a/442617) cites Ritt, which shows that the function in question is not expressible using radicals only. However, as [Kevin Casto comments](https://mathoverflow.net/questions/442600/can-the-positive-root-of-this-polynomial-be-expressed-elementarily#comment1142413_442617), this result does not immediately answer the question of whether the function is an EL function.


[Kevin Casto's answer](https://mathoverflow.net/a/442650) cites a result of mine to show that the answer to [a version of] Iosif Pinelis's question is negative, but conditional on Schanuel's conjecture. So it is natural to ask if we can avoid assuming Schanuel's conjecture.


The answer is yes. The reference I usually cite is *Abel's Theorem in Problems and Solutions*, by V. B. Alekseev, Kluwer, 2004. Specifically, see the Appendix by Khovanskii. There are various theorems in this appendix which yield the desired negative answer, but perhaps the most explicit statement can be found in Section A.8:



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> If an algebraic equation is not solvable by radicals then it remains non-solvable using the logarithm, the exponentials, and the other meromorphic functions in the complex plane.
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