Schanuel's conjecture

In mathematics, specifically transcendental number theory, Schanuel's conjecture is a conjecture made by Stephen Schanuel in the 1960s concerning the transcendence degree of certain field extensions of the rational numbers.

The conjecture is as follows:

The conjecture can be found in Lang (1966).

The conjecture, if proven, would generalize most known results in transcendental number theory. The special case where the numbers z₁,...,z_n are all algebraic is the Lindemann–Weierstrass theorem. If, on the other hand, the numbers are chosen so as to make exp(z₁),...,exp(z_n) all algebraic then one would prove that linearly independent logarithms of algebraic numbers are algebraically independent, a strengthening of Baker's theorem.

The Gelfond–Schneider theorem follows from this strengthened version of Baker's theorem, as does the currently unproven four exponentials conjecture.

Schanuel's conjecture, if proved, would also settle the algebraic nature of numbers such as e + π and e^e, and prove that e and π are algebraically independent simply by setting z₁ = 1 and z₂ = πi, and using Euler's identity.

Euler's identity states that e^πi + 1 = 0. If Schanuel's conjecture is true then this is, in some precise sense involving exponential rings, the only relation between e, π, and i over the complex numbers.

Although ostensibly a problem in number theory, the conjecture has implications in model theory as well. Angus Macintyre and Alex Wilkie, for example, proved that the theory of the real field with exponentiation, ℝ_exp, is decidable provided Schanuel's conjecture is true. In fact they only needed the real version of the conjecture, defined below, to prove this result, which would be a positive solution to Tarski's exponential function problem.

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