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:

Lindemann–Weierstrass theorem—if $α 1, ..., α n$ are algebraic numbers that are linearly independent over the rational numbers $\mathbb {Q}$ , then $e, ..., e$ are algebraically independent over $\mathbb {Q}$ .

In other words, the extension field $\mathbb {Q} (e^{\alpha _{1}},\dots ,e^{\alpha _{n}})$ has transcendence degree $n$ over $\mathbb {Q}$ .

An equivalent formulation from Baker 1990, Chapter 1, Theorem 1.4, is the following:

An equivalent formulation—If $α 1, ..., α n$ are distinct algebraic numbers, then the exponentials $e, ..., e$ are linearly independent over the algebraic numbers.

This equivalence transforms a linear relation over the algebraic numbers into an algebraic relation over $\mathbb {Q}$ 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 $e$ is transcendental for every non-zero algebraic number $α,$ thereby establishing that $π$ is transcendental (see below). Weierstrass proved the above more general statement in 1885.

The theorem, along with the Gelfond–Schneider theorem, is extended by Baker's theorem, and all of these would be further generalized by Schanuel's conjecture.