Lindemann-Weierstrass theorem

In mathematics, the Lindemann-Weierstrass theorem states that if α₁,...,α_n are algebraic numbers which are linearly independent over the rational numbers, then are algebraically independent over the algebraic numbers; in other words the set has transcendence degree n over . An equivalent formulation of the theorem is that if α₁,...,α_n are distinct algebraic numbers then are linearly independent over the algebraic numbers.

The theorem is named for Ferdinand von Lindemann, who proved the particular result that π is transcendental, and Karl Weierstraß.

Transcendence of e and π

The transcendence of e and π are direct corollaries of this theorem. Suppose α is a nonzero algebraic number; then {α} is a linearly independent set over the rationals, and therefore {e^α} has transcendence degree one over the rationals; or in other words e^α is transcendental. Using the other formulation we can argue that if {0, α} is a set of distinct algebraic numbers, then the set {e⁰, e^α} = {1, e^α} is linearly independent over the algebraic numbers, and so e^α is immediately seen to be transcendental. In particular, e^{1 = e is transcendental. Also, if β = eiα is transcendental, so are the real and imaginary parts of β, Re(β) = (β + β−1)/2 and Im(β) = (β − β−1)/2i, and hence cos(α) = Re(β) and sin(α) = Im(β) are also. Therefore, if π were algebraic, cos(π) = −1 and sin(π) = 0 would be transcendental, which proves by contradiction π is not algebraic, and hence is transcendental.}

p-adic conjecture

The p-adic Lindemann-Weierstrass conjecture is that this conjecture is also true for a p-adic analog: if α₁,...,α_n are a set of algebraic numbers linearly independent over the rationals such that | α_i | _p < 1 / p for some prime p, then the p-adic exponentials are algebraically independent transcendentals.