# 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.

## 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 [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
##### Navigation

Academic Kids Menu

• Art and Cultures
• Art (http://www.academickids.com/encyclopedia/index.php/Art)
• Architecture (http://www.academickids.com/encyclopedia/index.php/Architecture)
• Cultures (http://www.academickids.com/encyclopedia/index.php/Cultures)
• Music (http://www.academickids.com/encyclopedia/index.php/Music)
• Musical Instruments (http://academickids.com/encyclopedia/index.php/List_of_musical_instruments)
• Biographies (http://www.academickids.com/encyclopedia/index.php/Biographies)
• Clipart (http://www.academickids.com/encyclopedia/index.php/Clipart)
• Geography (http://www.academickids.com/encyclopedia/index.php/Geography)
• Countries of the World (http://www.academickids.com/encyclopedia/index.php/Countries)
• Maps (http://www.academickids.com/encyclopedia/index.php/Maps)
• Flags (http://www.academickids.com/encyclopedia/index.php/Flags)
• Continents (http://www.academickids.com/encyclopedia/index.php/Continents)
• History (http://www.academickids.com/encyclopedia/index.php/History)
• Ancient Civilizations (http://www.academickids.com/encyclopedia/index.php/Ancient_Civilizations)
• Industrial Revolution (http://www.academickids.com/encyclopedia/index.php/Industrial_Revolution)
• Middle Ages (http://www.academickids.com/encyclopedia/index.php/Middle_Ages)
• Prehistory (http://www.academickids.com/encyclopedia/index.php/Prehistory)
• Renaissance (http://www.academickids.com/encyclopedia/index.php/Renaissance)
• Timelines (http://www.academickids.com/encyclopedia/index.php/Timelines)
• United States (http://www.academickids.com/encyclopedia/index.php/United_States)
• Wars (http://www.academickids.com/encyclopedia/index.php/Wars)
• World History (http://www.academickids.com/encyclopedia/index.php/History_of_the_world)
• Human Body (http://www.academickids.com/encyclopedia/index.php/Human_Body)
• Mathematics (http://www.academickids.com/encyclopedia/index.php/Mathematics)
• Reference (http://www.academickids.com/encyclopedia/index.php/Reference)
• Science (http://www.academickids.com/encyclopedia/index.php/Science)
• Animals (http://www.academickids.com/encyclopedia/index.php/Animals)
• Aviation (http://www.academickids.com/encyclopedia/index.php/Aviation)
• Dinosaurs (http://www.academickids.com/encyclopedia/index.php/Dinosaurs)
• Earth (http://www.academickids.com/encyclopedia/index.php/Earth)
• Inventions (http://www.academickids.com/encyclopedia/index.php/Inventions)
• Physical Science (http://www.academickids.com/encyclopedia/index.php/Physical_Science)
• Plants (http://www.academickids.com/encyclopedia/index.php/Plants)
• Scientists (http://www.academickids.com/encyclopedia/index.php/Scientists)
• Social Studies (http://www.academickids.com/encyclopedia/index.php/Social_Studies)
• Anthropology (http://www.academickids.com/encyclopedia/index.php/Anthropology)
• Economics (http://www.academickids.com/encyclopedia/index.php/Economics)
• Government (http://www.academickids.com/encyclopedia/index.php/Government)
• Religion (http://www.academickids.com/encyclopedia/index.php/Religion)
• Holidays (http://www.academickids.com/encyclopedia/index.php/Holidays)
• Space and Astronomy
• Solar System (http://www.academickids.com/encyclopedia/index.php/Solar_System)
• Planets (http://www.academickids.com/encyclopedia/index.php/Planets)
• Sports (http://www.academickids.com/encyclopedia/index.php/Sports)
• Timelines (http://www.academickids.com/encyclopedia/index.php/Timelines)
• Weather (http://www.academickids.com/encyclopedia/index.php/Weather)
• US States (http://www.academickids.com/encyclopedia/index.php/US_States)

Information

• Home Page (http://academickids.com/encyclopedia/index.php)
• Contact Us (http://www.academickids.com/encyclopedia/index.php/Contactus)

• Clip Art (http://classroomclipart.com)