Lindemann–Weierstrass theorem
On algebraic independence of exponentials of linearly independent algebraic numbers over Q / From Wikipedia, the free encyclopedia
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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:
Lindemann–Weierstrass theorem — if α1, ..., αn are algebraic numbers that are linearly independent over the rational numbers , then eα1, ..., eαn are algebraically independent over .
In other words, the extension field has transcendence degree n over .
An equivalent formulation (Baker 1990, Chapter 1, Theorem 1.4), is the following:
An equivalent formulation — If α1, ..., αn are distinct algebraic numbers, then the exponentials eα1, ..., eαn are linearly independent over the algebraic numbers.
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.