Schanuel's conjecture
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In mathematics, specifically transcendence theory, Schanuel's conjecture is the following statement:
- Given any n complex numbers z1,...,zn which are linearly independent over the rational numbers Q, then the extension field Q(z1,...,zn,exp(z1),...,exp(zn)) has transcendence 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)[1]. No proof is known.
The conjecture, if proven, would imply the Lindemann-Weierstrass theorem, the Gelfond-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[2] is the following statement:
- Suppose F is a 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 x1,...,xn of F which are linearly independent over Q, the extension field Q(x1,...,xn,e(x1),...,e(xn)) has transcendence degree at least n over Q. Then there exists a field homomorphism h : E → 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 by James Ax in 1971.[3] It states:
- Given any n formal power series f1,...,fn in tC[[t]] which are linearly independent over Q, then the field extension C(t,f1,...,fn,exp(f1),...,exp(fn)) has transcendence degree at least n over C(t).
[edit] References
- ^ Serge Lang. Introduction to Transcendental Numbers. Addison-Wesley 1966. Pages 30-31
- ^ Scott W. Williams. Million Bucks Problems
- ^ James Ax. On Schanuel's conjectures. Annals of Mathematics(2) 93, 1971, pages 252-268.
