lindemannweierstrass

The Lindemann–Weierstrass theorem is a foundational result in transcendental number theory, named after Ferdinand von Lindemann and Karl Weierstrass. It generalizes Lindemann’s earlier work showing the transcendence of certain exponentials and gives a strong statement about the linear independence of exponentials at algebraic arguments.

Statement: If α1, …, αn are distinct algebraic numbers, then e^{α1}, …, e^{αn} are linearly independent over the field of algebraic numbers. A key consequence is that for any nonzero algebraic number α, e^{α} is transcendental. The theorem also yields that numbers such as e^{α1}, …, e^{αn} cannot satisfy nontrivial algebraic relations with algebraic coefficients, unless all coefficients are zero.

History and significance: The result was proved by Lindemann in 1882, building on Hermite’s earlier transcendence proof for e. Weierstrass contributed refinements and a broader formulation, leading to the combined name Lindemann–Weierstrass. The theorem is central to transcendence theory, enabling proofs of the transcendence of many individual constants (such as e and π, via corollaries) and providing powerful tools for establishing algebraic independence of exponential values.

Applications and consequences: The theorem implies the transcendence of e and, through classical corollaries, the transcendence of π. It underpins numerous results about exponential values at algebraic arguments and informs broader questions about which numbers can be algebraic versus transcendental, as well as linear and algebraic independence among exponential expressions.