Schanuel's conjecture and real root of $x+e^x=0$ Schanuel's conjecture states:


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*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 (The American Mathematical Monthly, Vol. 106, No. 5. (May, 1999), pp. 440-448; MSN), 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:


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*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 $R\in\mathbb{E}$?
 A: 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 and in Risch, R. H.: Algebraic Properties of the Elementary Functions of Analysis. Amer. J.
Math. 101 (1979) (4) 743-759. 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.
