Lindemann-Weierstrass theorem

The Lindemann-Weierstrass theorem is a theorem in mathematics that is very useful in establishing the transcendence of numbers. The theorem states:

If α₁,...,α_n are distinct algebraic numbers, and β₁,...,β_n are any nonzero algebraic numbers, then

The transcendence of e and &pi are direct corollaries of this theorem. To show the transcendence of e, note that if e were algebraic, there would exist rational_numbers β₀,...,β_n, not all zero, such that

Since every rational number is algebraic, this violates the Lindemann-Weierstrass theorem, and so e must be transcendental.

To show the transcendence of π, suppose that π was algebraic. Since the set of all algebraic numbers forms a field, this implies that πi and 2πi are also algebraic. Taking β₁ = β₂ = 1, α₁ = πi, α₂ = 2πi, the Lindemann-Weierstrass theorem gives us the equation (see Euler's formula)

and this contradiction establishes the transcendence of π.

The theorem is named for Carl Louis Ferdinand von Lindemann and Karl Weierstraß