The Lindemann-Weierstrass Theorem is a mathematical theorem that states that for any non-zero algebraic number a, the number ea is transcendental. This means that e raised to any non-zero algebraic power will always be a transcendental number, meaning it cannot be expressed as the root of a polynomial equation with rational coefficients.
The theorem was first proven by the German mathematician Ferdinand von Lindemann in 1882. However, it was independently discovered and proven by the German mathematician Karl Weierstrass in 1885.
The theorem has several important implications in mathematics. It provides a proof for the transcendence of e, which was previously only known through intuition and conjecture. It also has applications in other areas of mathematics, such as the study of complex numbers and algebraic geometry.
The Lindemann-Weierstrass Theorem is closely related to the famous problem of squaring the circle, which involves constructing a square with the same area as a given circle using only a compass and a straightedge. The theorem proves that this is impossible, as it shows that the number pi (which is the ratio of a circle's circumference to its diameter) is a transcendental number. This means that it cannot be constructed using only rational numbers and basic geometric operations.
Yes, the theorem has been used to prove the transcendence of other important mathematical constants, such as the number pi2 and the Euler-Mascheroni constant. It has also been applied in the study of elliptic functions and modular forms, and has connections to the Riemann hypothesis in number theory.