- Schanuel's conjecture
In

mathematics , specificallytranscendence theory ,**Schanuel's conjecture**is the following statement::Given any "n"complex number s "z"_{1},...,"z"_{"n"}which are linearly independent over therational number s**Q**, the extension field**Q**("z"_{1},...,"z"_{"n"},exp("z"_{1}),...,exp("z"_{"n"})) hastranscendence degree of at least "n" over**Q**.The conjecture was formulated by

Stephen Schanuel in the early 1960s and can be found in (Lang 1966) [] . No proof is known.Serge Lang . "Introduction to Transcendental Numbers." Addison-Wesley 1966. Pages 30-31The conjecture, if proven, would imply the

Lindemann-Weierstrass theorem , theGelfond-Schneider theorem and several other results about transcendence properties of the exponential function, as well as the (as yet unproven)algebraic independence of π and "e".The

**converse Schanuel conjecture**[*Scott W. Williams. [*] is the following statement: :Suppose "F" is a*http://www.math.buffalo.edu/~sww/0papers/million.buck.problems.mi.pdf Million Bucks Problems*]countable field with characteristic 0, and "e" : "F" → "F" is a homomorphism from the additive group ("F",+) to the multiplicative group ("F",·) whose kernel is cyclic. Suppose further that for any "n" elements "x"_{1},...,"x"_{"n"}of "F" which are linearly independent over**Q**, the extension field**Q**("x"_{1},...,"x"_{"n"},"e"("x"_{1}),...,"e"("x"_{"n"})) has transcendence degree at least "n" over**Q**. Then there exists a field homomorphism "h" : "F" →**C**such that "h"("e"("x"))=exp("h"("x")) for all "x" in "F".A version of Schanuel's conjecture for

formal power series , also by Schanuel, was proven byJames Ax in 1971. [*James Ax. On Schanuel's conjectures. "*] It states::Given any "n" formal power series "f"Annals of Mathematics "(2) 93, 1971, pages 252-268._{1},...,"f"_{"n"}in*t***C** "t"which are linearly independent over **Q**, then the field extension**C**("t","f"_{1},...,"f"_{"n"},exp("f"_{1}),...,exp("f"_{"n"})) has transcendence degree at least "n" over**C**("t").Boris Zilber [

*Boris Zilber. Pseudo-exponentiation on algebraically closed fields of characteristic zero,*] constructed an axiomatization of pseudo-exponentiation in algebraically closed fields of characteristic zero. Using Hrushovski's construction, he proved that the theory is satisfiable, and categorical in all uncountable powers. The resulting theory has a unique model of cardinality of the continuum. If Schanuel's conjecture is true, then (C,+,x,exp) is the unique model of this cardinality. Conversely, the Hrushovski inequality formulated in these models is Schanuel's conjecture. This doesn't prove Schanuel's conjecture, however, since we don't know that the unique model is (C,+,x,exp).Annals of Pure and Applied Logic , (132), 2004, 1, pp 67-95**ee also***Schanuel's conjecture implies a positive solution to

Tarski's exponential function problem **References**

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