In transcendental number theory, Schanuel's conjecture is a conjecture about the transcendence degree of certain field extensions of the rational numbers, which would establish the transcendence of a large class of numbers for which this is currently unknown. It is due to Stephen Schanuel and was published by Serge Lang in 1966. This description is adapted from Wikipedia contributors under CC BY-SA 4.0; changes were made. https://creativecommons.org/licenses/by-sa/4.0/
Facts
StatementFor any n complex numbers z1,...,zn that are linearly independent over the rational numbers, the field extension Q(z1,...,zn, e^z1,...,e^zn) has transcendence degree at least n over Q. 1 Proposed Year1966 is the year Serge Lang published the conjecture; it is attributed to Stephen Schanuel. Progress Toward ResolutionOpen. The analogue for formal power series was proved by James Ax in 1971, and it is known that the theory of the real field with exponentiation is decidable provided Schanuel's conjecture is true (Macintyre and Wilkie). 1 Classification
Resolution Status Prize Status
Prize Status (category) Sources
1. Schanuel's Conjecture (Wikipedia)
Wikimedia FoundationLead section
It is due to Stephen Schanuel and was published by Serge Lang in 1966
Statement section
has transcendence degree at least n
Related results section, Ax sentence
A version of Schanuel's conjecture for formal power series, also by Schanuel, was proven by James Ax in 1971.
Related results section, Macintyre and Wilkie sentence
is decidable provided Schanuel's conjecture is true
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