During the Arizona Winter School 2008 (held in Tucson, AZ) we worked on the following problems: a) (Expanding a remark by S. Lang). Define $E_0 = overline{mathbb{Q}}$ Inductively, for $n geq 1$, define $E_n$ as the algebraic closure of the field generated over $E_{n-1}$ by the numbers $exp(x)=e^x$, where $x$ ranges over $E_{n-1}$. Let $E$ be the union of $E_n$, $n geq 0$. Show that Schanuels Conjecture implies that the numbers $pi, log pi, log log pi, log log log pi, ldots $ are algebraically independent over $E$. b) Try to get a (conjectural) generalization involving the field $L$ defined as follows. Define $L_0 = overline{mathbb{Q}}$. Inductively, for $n geq 1$, define $L_n$ as the algebraic closure of the field generated over $L_{n-1}$ by the numbers $y$, where $y$ ranges over the set of complex numbers such that $e^yin L_{n-1}$. Let $L$ be the union of $L_n$, $n geq 0$. We were able to prove that Schanuels Conjecture implies $E$ and $L$ are linearly disjoint over $overline{mathbb{Q}}$.
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