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Lindemann-weierstrass TheoremIn mathematics, the Lindemann-Weierstrass theorem states that if α1,...,αn 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 transcendency degree n over . An equivalent formulation of the theorem is that if α1,...,αn 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 Weierstra. 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 {eα} has transcendence degree one over the rationals; or in other words eα is transcendental. Using the other formulation we can argue that if {0, α} is a set of distinct algebraic numbers, then the set {e0, eα} = {1, eα} is linearly independent over the algebraic numbers, and so eα is immediately seen to be transcendental. In particular, e1 = 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,...,αn are a set of algebraic numbers linearly independent over the rationals such that for some prime p, then the p-adic exponentials are algebraically independent transcendentals.
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