Schanuel's conjecture states:


* If $\alpha\_1,\alpha\_2,...,\alpha\_n$ are complex numbers linearly independent over $\mathbb{Q}$, then the transcendence degree of the field $\mathbb{Q}(\alpha\_1,e^{\alpha\_1},\alpha\_2,e^{\alpha\_2},...,\alpha\_n,e^{\alpha\_n})$ over $\mathbb{Q}$ is at least $n$.


In [What is a closed-form number](https://www.jstor.org/stable/2589148) (The American Mathematical Monthly, Vol. 106, No. 5. (May, 1999), pp. 440-448; [MSN](https://www.ams.org/mathscinet-getitem?mr=1699262)), Timothy Y. Chow states the following conjecture: 


A) The real root $R$ of the equation $x+e^x=0$ is not in $\mathbb{E}$ (having defined $\mathbb{E}$ as the field of all ‘logarithmic exponential numbers’, which is the intersection of all subfields in $\mathbb{C}$ that are closed under $\exp$ and $\log$).


He also states:


* Schanuel's conjecture implies conjecture A).


My question is: if Schanuel's conjecture isn't true, does this mean we can find the solution $R$ of $x+e^x=0$ with $R\in\mathbb{E}$?



Schanuel's conjecture and real root of $x+e^x=0$
My answer is not the whole answer.


$R=-W(1)$, where $W$ is the Lambert W function.


$H(x)=x+e^{x}=0$, $H$ is an elementary function and the inverse of $H$ is $H^{-1}$ with $H^{-1}(x)=-W(e^{x})+x$.


Let us assume Schanuel's conjecture is not true and $-W(1)$ is an elementary number. That does not mean the equation $x+e^{x}=0$ can be solved **by rearraning the equation according to $x$ by applying only elementary functions that we can read off from the equation** and we get $R$ on this way as an elementary expression of a rational number. Solving an equation **on this way** means applying the inverse of $H$. Solvability of equations **on this way** is therefore related to the problem of invertibility of elementary functions by elementary functions. This problem was solved in [Ritt, J. F.: Elementary functions and their inverses. Trans. Amer. Math. Soc. 27 (1925) (1) 68-90](http://www.ams.org/journals/tran/1925-027-01/home.html) and in [Risch, R. H.: Algebraic Properties of the Elementary Functions of Analysis. Amer. J.
Math. 101 (1979) (4) 743-759](https://www.jstor.org/stable/2373917?seq=1#page_scan_tab_contents). That Lambert W is a non-elementary function follows already from the theorem of J. F. Ritt and Lindemann-Weierstrass theorem.


One would have to show by other methods that $-W(1)$ is an elementary number.


