I just read an article of Ram Murty about transcendence of special values of Lfunctions, and it seems that Schanuel's conjecture plays a crucial role in it. So given a positive integer $n$, let's define the set $\mathcal{L}_{n}$ in the following way: $A$ is an element of $\mathcal{L}_{n}$ if and only if $A$ is a set of $n$ complex numbers linearly independent over $\mathbb{Q}$. Now let's denote $E(A)$ the set $\{e^{\alpha_{i}},\alpha_{i}\in A\}$.
My question is: Is it true that there exists a positive integer $k_n$ depending only on $n$ such that for all $\sigma\in\frak{S}(\mathcal{L}_{n})$ the transcendence degree of $\mathbb{Q}(\sigma(A),E(\sigma(A)))$ is $k_n$? If so, proving Schanuel's conjecture would pertain to showing that $k_n\geqslant n$.
Thanks in advance.
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