Lindemann–Weierstrass theorem
In transcendental number theory, the Lindemann–Weierstrass theorem is a result that is very useful in establishing the transcendence of numbers. It states the following:
In other words, the extension field has transcendence degree n over .
An equivalent formulation (Baker 1990, Chapter 1, Theorem 1.4), is the following:
This equivalence transforms a linear relation over the algebraic numbers into an algebraic relation over by using the fact that a symmetric polynomial whose arguments are all conjugates of one another gives a rational number.
The theorem is named for Ferdinand von Lindemann and Karl Weierstrass. Lindemann proved in 1882 that eα is transcendental for every non-zero algebraic number α, thereby establishing that π is transcendental (see below).[1] Weierstrass proved the above more general statement in 1885.[2]
The theorem, along with the Gelfond–Schneider theorem, is extended by Baker's theorem,[3] and all of these would be further generalized by Schanuel's conjecture.
Naming convention[edit]
The theorem is also known variously as the Hermite–Lindemann theorem and the Hermite–Lindemann–Weierstrass theorem. Charles Hermite first proved the simpler theorem where the αi exponents are required to be rational integers and linear independence is only assured over the rational integers,[4][5] a result sometimes referred to as Hermite's theorem.[6] Although that appears to be a special case of the above theorem, the general result can be reduced to this simpler case. Lindemann was the first to allow algebraic numbers into Hermite's work in 1882.[1] Shortly afterwards Weierstrass obtained the full result,[2] and further simplifications have been made by several mathematicians, most notably by David Hilbert[7] and Paul Gordan.[8]
Related result[edit]
A variant of Lindemann–Weierstrass theorem in which the algebraic numbers are replaced by the transcendental Liouville numbers (or in general, the U numbers) is also known.[12]