Some consequences of Schanuel's Conjecture

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During the Arizona Winter School 2008 (held in Tucson, AZ) we worked on the following problems: a) (Expanding a remark by S. Lang). Define E₀ = Q Inductively, for n ≥ 1, define Eₙ as the algebraic closure of the field generated over Eₙ₋₁ by the numbers (x)=eˣ, where x ranges over Eₙ₋₁. Let E be the union of Eₙ, n ≥ 0. Show that Schanuel's Conjecture implies that the numbers π, π, π, π, are algebraically independent over E. b) Try to get a (conjectural) generalization involving the field L defined as follows. Define L₀ = Q. Inductively, for n ≥ 1, define Lₙ as the algebraic closure of the field generated over Lₙ₋₁ by the numbers y, where y ranges over the set of complex numbers such that eʸ∈ Lₙ₋₁. Let L be the union of Lₙ, n ≥ 0. We were able to prove that Schanuel's Conjecture implies E and L are linearly disjoint over Q.