Сандық теориядағы Шанюэль болжамы

Conjecture on the transcendence degree of field extensions to the rational numbers

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, if proven, would generalize most known results in transcendental number theory. The special case where the numbers z1, ,zn are all algebraic is the Lindemann–Weierstrass theorem. If, on the other hand, the numbers are chosen so as to make exp(z1), ,exp(zn) 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 whether numbers such as e +  and ee are algebraic or transcendental, and prove that e and are algebraically independent simply by setting z1 = 1 and z2 = i, and using Euler's identity. Euler's identity states that ei + 1 = 0. If Schanuel's conjecture is true then this is, in some precise sense involving exponential rings, the only relation between e, , 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, 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.

While a proof of Schanuel's conjecture seems a long way off, connections with model theory have prompted a surge of research on the conjecture. In 2004, Boris Zilber systematically constructed exponential fields Kexp that are algebraically closed and of characteristic zero, and such that one of these fields exists for each uncountable cardinality. He axiomatised these fields and, using Hrushovski's construction and techniques inspired by work of Shelah on categoricity in infinitary logics, proved that this theory of "pseudo exponentiation" has a unique model in each uncountable cardinal. Schanuel's conjecture is part of this axiomatisation, and so the natural conjecture that the unique model of cardinality continuum is actually isomorphic to the complex exponential field implies Schanuel's conjecture. In fact, Zilber showed that this conjecture holds if and only if both Schanuel's conjecture and another unproven condition on the complex exponentiation field, which Zilber calls exponential algebraic closedness, hold. As this construction can also give models with counterexamples of Schanuel's conjecture, this method cannot prove Schanuel's conjecture.