Schanuel's guess

The Schanuel's conjecture is still unproved mathematical statement about the transcendence degrees of certain field extensions of the body of rational numbers . This conjecture therefore belongs to the field of transcendence studies of algebra and algebraic number theory . It was formulated in the 1960s by Stephen Schanuel , after whom it is also named. ${\displaystyle \mathbb {Q} }$

The presumption

Let be a set of different complex numbers that are over linearly independent . ${\displaystyle \lbrace z_{1},z_{2},\ldots z_{n}\rbrace \subset \mathbb {C} }$ ${\displaystyle n}$ ${\displaystyle \mathbb {Q} }$

Then the expansion body has at least the degree of transcendence . ${\displaystyle T=\mathbb {Q} \left(z_{1},z_{2},\ldots z_{n},e^{z_{1}},e^{z_{2}},\ldots e^{z_{n}}\right)}$ ${\displaystyle \mathbb {Q} }$ ${\displaystyle n}$

The conjecture is still unproven to this day (July 2015).

Inferences

Schanuel's conjecture encompasses most of the known and proven theorems and some well-known conjectures about the transcendence of numbers as a special case.

• The set of Lindemann-Weierstrass arises in the special case that the amount of only algebraic numbers there. Then the degree of transcendence is exact . ${\displaystyle \lbrace z_{1},z_{2},\ldots z_{n}\rbrace \subset \mathbb {C} }$ ${\displaystyle T}$ ${\displaystyle n}$
• If, on the other hand, one chooses these numbers in such a way that they are a set of algebraic and -linearly independent numbers, then a (as yet unproven) generalization of a theorem by Alan Baker results . ${\displaystyle n}$ ${\displaystyle \lbrace e^{z_{1}},e^{z_{2}},\ldots e^{z_{n}}\rbrace \subset \mathbb {C} }$ ${\displaystyle n}$ ${\displaystyle \mathbb {Q} }$

• From this stronger version of Baker's theorem, the Gelfond-Schneider Theorem (proven in the 1930s) would follow.

• Schanuel's conjecture would also show that combinations are like transcendent and that algebraically is independent. ${\displaystyle e+\pi ,e^{e},\pi ^{\pi }}$ ${\displaystyle \lbrace e,\pi \rbrace }$
• From Euler's formula it follows that it holds. Should Schanuel's conjecture be correct, then this would essentially be the only relation of this kind between the numbers over the whole numbers in a more precise sense . ${\displaystyle e^{i\pi }+1=0}$ ${\displaystyle e,\pi ,i}$

• Angus Macintyre showed as early as 1991 that it follows from Schanuel's conjecture that there are no such “unexpected” exponential-algebraic relations over the whole numbers.

Reversal of the assumption

The following statement is called the reverse of Schanuel's conjecture :

Let be a countable field with the characteristic 0, a group homomorphism whose core is a cyclic group . It also applies that for more than linearly independent elements of the extension field most the transcendence degree always on has. Then there is a body automorphism so that applies to all . ${\displaystyle L}$ ${\displaystyle \epsilon :(L,+)\rightarrow (L^{*},\cdot )}$ ${\displaystyle n}$ ${\displaystyle \mathbb {Q} }$ ${\displaystyle z_{1},z_{2},\ldots z_{n}\in L}$ ${\displaystyle T=\mathbb {Q} \left(z_{1},z_{2},\ldots z_{n},\epsilon (z_{1}),\epsilon (z_{2}),\ldots \epsilon (z_{n})\right)}$ ${\displaystyle n}$ ${\displaystyle \mathbb {Q} }$ ${\displaystyle h:L\rightarrow \mathbb {C} }$ ${\displaystyle h(\epsilon (z))=e^{h(z)}}$ ${\displaystyle z\in L}$

See also

• Hilbert's seventh problem in Hilbert's list of 23 math problems .

literature

• Alan Baker : The theory of linear forms in logarithms. In: Alan Baker, David W. Masser (Eds.): Transcendence Theory. Advances and Applications. Proceedings of a Conference held at Cambridge in 1976. Academic Press, London 1977, ISBN 0-12-074350-7 , pp. 1-27.
• Alan Baker: Transcendental number theory . Reissued as a paperback with updated material. 2nd Edition. Cambridge University Press, Cambridge u. a. 1990, ISBN 0-521-39791-X .
• Gregory Chudnovsky : On the Way to Schanuel's Conjecture. In: Gregory Chudnovsky: Contributions to the Theory of transcendental Numbers (= Mathematical Surveys and Monographs. Volume 19). American Mathematical Society, Providence RI 1984, ISBN 0-8218-1500-8 , pp. 145-176.
• Serge Lang : Introduction to Transcendental Numbers . Addison-Wesley, Reading MA et al. a. 1966.
• Angus Macintyre : Schanuel's Conjecture and Free Exponential Rings . In: Annals of Pure and Applied Logic . tape 51 , no. 3 , 1991, ISSN  0168-0072 , pp. 241-246 , doi : 10.1016 / 0168-0072 (91) 90017-G .
• David Marker: Model Theory and Exponentiation . In: Notices of the American Mathematical Society . tape 43 , 1996, ISSN  1088-9477 , pp. 753-759 ( ams.org [PDF; 205 kB ]).
• Giuseppina Terzo: Some consequences of Schanuel's conjecture in exponential rings . In: Communications in Algebra . tape 36 , no. 3 , 2008, ISSN  0092-7872 , p. 1171-1189 , doi : 10.1080 / 00927870701410694 .

Web links

• Proof of the transcendence of e and π in the evidence archive .
• Eric W. Weisstein : Schanuel's Conjecture . In: MathWorld (English).

Individual evidence

1. ↑ Lang (1966), pp. 30f
2. ↑ Baker (1977)
3. ↑ Terzo (2008)
4. ↑ Mcintyre (1991)
5. ↑ Scott W. Williams: Million Bucks Problems (PDF; 17 kB)