In mathematics, specifically transcendental number theory, Schanuel's conjecture is a conjecture about the transcendence degree of certain field extensions of the rational numbers Q {\displaystyle \mathbb {Q} } , 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.
Statement Schanuel's conjecture can be given as follows:
Consequences Schanuel's conjecture, if proven, would generalize most known results in transcendental number theory and establish a large class of numbers transcendental. Special cases of Schanuel's conjecture include:
Lindemann–Weierstrass theorem
Considering Schanuel's conjecture for only n = 1 {\displaystyle n=1} gives that for a nonzero complex number z {\displaystyle z} , at least one of the numbers z {\displaystyle z} and e z {\displaystyle e^{z}} must be transcendental. This was proved by Ferdinand von Lindemann in 1882. If the numbers z 1 , . . . , z n {\displaystyle z_{1},...,z_{n}} are taken to be all algebraic and linearly independent over Q {\displaystyle \mathbb {Q} } then the e z 1 , . . . , e z n {\displaystyle e^{z_{1}},...,e^{z_{n}}} result to be transcendental and algebraically independent over Q {\displaystyle \mathbb {Q} } . The first proof for this more general result was given by Carl Weierstrass in 1885. This so-called Lindemann–Weierstrass theorem implies the transcendence of the numbers e and π. It also follows that for algebraic numbers α {\displaystyle \alpha } not equal to 0 or 1, both e α {\displaystyle e^{\alpha }} and ln ( α ) {\displaystyle \ln(\alpha )} are transcendental. It further gives the transcendence of the trigonometric functions at nonzero algebraic values.
Baker's theorem
Another special case was proved by Alan Baker in 1966: If complex numbers λ 1 , . . . , λ n {\displaystyle \lambda _{1},...,\lambda _{n}} are chosen to be linearly independent over the rational numbers Q {\displaystyle \mathbb {Q} } such that e λ 1 , . . . , e λ n {\displaystyle e^{\lambda _{1}},...,e^{\lambda _{n}}} are algebraic, then λ 1 , . . . , λ n {\displaystyle \lambda _{1},...,\lambda _{n}} are also linearly independent over the algebraic numbers Q ¯ {\displaystyle \mathbb {\overline {Q}} } . Schanuel's conjecture would strengthen this result, implying that λ 1 , . . . , λ n {\displaystyle \lambda _{1},...,\lambda _{n}} would also be algebraically independent over Q {\displaystyle \mathbb {Q} } (and equivalently over Q ¯ {\displaystyle \mathbb {\overline {Q}} } ).
Gelfond–Schneider theorem
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