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 that if α₁, ..., α_n are algebraic numbers which are linearly independent over the rational numbers ℚ, then e^α₁, ..., e^α_n are algebraically independent over ℚ; in other words the extension field ℚ(e^α₁, ..., e^α_n) has transcendence degree n over ℚ.

An equivalent formulation (Baker 1975, Chapter 1, Theorem 1.4), is the following: If α₁, ..., α_n are distinct algebraic numbers, then the exponentials e^α₁, ..., e^α_n are linearly independent over the algebraic numbers. This equivalence transforms a linear relation over the algebraic numbers into an algebraic relation over 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^α 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.

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