Editing Lindemann–Weierstrass theorem (section)

{{short description|On algebraic independence of exponentials of linearly independent algebraic numbers over Q}}
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In [[transcendental number theory]]<!--[[mathematics]]-->, the '''Lindemann–Weierstrass theorem''' is a result that is very useful in establishing the [[transcendental number|transcendence]] of numbers. It states the following: {{math_theorem|name=Lindemann–Weierstrass theorem|if {{math|α<sub>1</sub>,&nbsp;...,&nbsp;α<sub>''n''</sub>}} are [[algebraic number]]s that are [[linearly independent]] over the [[rational number]]s <math>\mathbb{Q}</math>, then {{math|''e''<sup>α<sub>1</sub></sup>,&nbsp;...,&nbsp;''e''<sup>α<sub>''n''</sub></sup>}} are [[algebraically independent]] over <math>\mathbb{Q}</math>.}} 
In other words, the [[extension field]] <math>\mathbb{Q}(e^{\alpha_1}, \dots, e^{\alpha_n})</math> has [[transcendence degree]] {{math|''n''}} over <math>\mathbb{Q}</math>.

An equivalent formulation from {{Harvnb|Baker|1990|loc=Chapter 1, Theorem 1.4}}, is the following: {{math_theorem|name=An equivalent formulation|If {{math| α<sub>1</sub>,&nbsp;...,&nbsp;α<sub>''n''</sub> }} are distinct algebraic numbers, then the exponentials {{math|''e''<sup>α<sub>1</sub></sup>,&nbsp;...,&nbsp;''e''<sup>α<sub>''n''</sub></sup>}} are linearly independent over the algebraic numbers.}}  This equivalence transforms a linear relation over the algebraic numbers into an algebraic relation over <math>\mathbb{Q}</math> by using the fact that a [[symmetric polynomial]] whose arguments are all [[algebraic conjugate|conjugates]] of one another gives a rational number.

The theorem is named for [[Ferdinand von Lindemann]] and [[Karl Weierstrass]]. Lindemann proved in 1882 that {{math|''e''<sup>α</sup>}} is transcendental for every non-zero algebraic number {{math|α,}} thereby establishing that {{pi}} is transcendental (see below).<ref name="Lindemann1882a" /> Weierstrass proved the above more general statement in 1885.<ref name="Weierstrass1885" />

The theorem, along with the [[Gelfond–Schneider theorem]], is extended by [[Baker's theorem]],<ref>{{Harvnb|Murty|Rath|2014}}</ref> and all of these would be further generalized by [[Schanuel's conjecture]].