Schanuel's conjecture

The conjecture of Schanuel is an up to now unproven mathematical statement about the transcendence degree of certain field extensions of the field of rational numbers. Thus, this assumption belongs to the realm of transcendence studies algebra and algebraic number theory. It was formulated in the 1960s by Stephen Schanuel, after whom it is named.

The assumption

Is a set of different complex numbers, which are linearly independent.

Then, the extension field over at least the transcendence degree.

The assumption is unproven to date ( October 2013 ).

Conclusions

The conjecture of Schanuel includes the most well-known and proven principles 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 set consists of only algebraic numbers. Then the transcendence degree of course is accurate.
• If one chooses the other hand, these figures so that a set of algebraic and linearly independent numbers is then given a (so far unproven ) generalization of a theorem by Alan Baker.

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

• The conjecture of Schanuel would also show that combinations are as transcendent and that is algebraically independent.
• From Euler's formula it follows that applies. If the conjecture of Schanuel true, then this would be the only relation between the numbers of this type over the integers in a sense präzisierbaren substantially.

• Angus Macintyre showed in 1991 that follows from the conjecture of Schanuel that there are no such "unexpected" exponential - algebraic relations over the integers.

Reversal of the presumption

As a reversal of the presumption of Schanuel the following statement is called:

Be a countable body with the characteristic 0, a group homomorphism whose kernel is a cyclic group. It is true also that there is always at most the transcendence degree over for linearly independent elements of the extension field over. Then there is a Körperautomorphismus such that for all.