# Lindemann-Weierstrass theorem

In mathematics, the Lindemann-Weierstrass theorem states that if α1,...,αn are algebraic numbers which are linearly independent over the rational numbers, then [itex]e^{\alpha_1} \cdots e^{\alpha_n}[itex] are algebraically independent over the algebraic numbers; in other words the set [itex]{e^{\alpha_1} \cdots e^{\alpha_n}}[itex] has transcendence degree n over [itex]\Bbb{Q}[itex]. An equivalent formulation of the theorem is the following one: If α1,...,αn are algebraic numbers, linearly independent over the rationals (and therefore necessarily distinct), then all the different monomial products [itex]e^{m_1\alpha_1} \cdots e^{m_n\alpha_n}[itex] with integer coefficients mi are linearly independent over the algebraic numbers. In reference [1] below the theorem is stated in the following form: If α1,...,αn are distinct algebraic numbers, then the exponentials [itex]e^{\alpha_1},\ldots,e^{\alpha_n}[itex] 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 {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</sub> = 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.

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 [itex]|\alpha_i|_p < 1/p[itex] for some prime p, then the p-adic exponentials [itex]e^{\alpha_1} \cdots e^{\alpha_n}[itex] are algebraically independent transcendentals.

## References

1. Alan Baker, Transcendental Number Theory, Cambridge University Press, 1975, chapter 1, Theorem 1.4. ISBN: 0 521 39791.de:Satz von Lindemann-Weierstrass

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