Lindemann–Weierstrass theorem

In transcendental number theory, the Lindemann–Weierstrass theorem is a result that is very useful in establishing the transcendence of numbers. It states the following. Lindemann–Weierstrass theorem — if α1, ..., αn are algebraic numbers that are linearly independent over the rational numbers ℚ, then eα1, ..., eαn are algebraically independent over ℚ.An equivalent formulation — If α1, ..., αn are distinct algebraic numbers, then the exponentials eα1, ..., eαn are linearly independent over the algebraic numbers.p-adic Lindemann–Weierstrass Conjecture. — Suppose p is some prime number and α1, ..., αn are p-adic numbers which are algebraic and linearly independent over ℚ, such that | αi |p < 1/p for all i; then the p-adic exponentials expp(α1), . . . , expp(αn) are p-adic numbers that are algebraically independent over ℚ. Modular conjecture — Let q1, ..., qn be non-zero algebraic numbers in the complex unit disc such that the 3n numbersLindemann–Weierstrass Theorem (Baker's reformulation). — If a1, ..., an are algebraic numbers, and α1, ..., αn are distinct algebraic numbers, thenLemma A. — Let c(1), ..., c(r) be integers and, for every k between 1 and r, let {γ(k)1, ..., γ(k)m(k)} be the roots of a non-zero polynomial with integer coefficients T k ( x ) {displaystyle T_{k}(x)} . If γ(k)i ≠ γ(u)v whenever (k, i) ≠ (u, v), thenLemma B. — If b(1), ..., b(n) are integers and γ(1), ..., γ(n), are distinct algebraic numbers, then In transcendental number theory, the Lindemann–Weierstrass theorem is a result that is very useful in establishing the transcendence of numbers. It states the following. In other words the extension field ℚ(eα1, ..., eαn) has transcendence degree n over ℚ. An equivalent formulation (Baker 1990, Chapter 1, Theorem 1.4), is the following. This equivalence transforms a linear relation over the algebraic numbers into an algebraic relation over ℚ: by using the fact that a symmetric polynomial whose arguments are all conjugates of one another gives a rational number. The theorem is named for Ferdinand von Lindemann and Karl Weierstrass. Lindemann proved in 1882 that eα is transcendental for every non-zero algebraic number α, thereby establishing that π is transcendental (see below). Weierstrass proved the above more general statement in 1885. The theorem, along with the Gelfond–Schneider theorem, is extended by Baker's theorem, and all of these are further generalized by Schanuel's conjecture. The theorem is also known variously as the Hermite–Lindemann theorem and the Hermite–Lindemann–Weierstrass theorem. Charles Hermite first proved the simpler theorem where the αi exponents are required to be rational integers and linear independence is only assured over the rational integers, a result sometimes referred to as Hermite's theorem. Although apparently a rather special case of the above theorem, the general result can be reduced to this simpler case. Lindemann was the first to allow algebraic numbers into Hermite's work in 1882. Shortly afterwards Weierstrass obtained the full result, and further simplifications have been made by several mathematicians, most notably by David Hilbert and Paul Gordan. The transcendence of e and π are direct corollaries of this theorem. Suppose α is a non zero algebraic number; then {α} is a linearly independent set over the rationals, and therefore by the first formulation of the theorem {eα} is an algebraically independent set; or in other words eα is transcendental. In particular, e1 = e is transcendental. (A more elementary proof that e is transcendental is outlined in the article on transcendental numbers.)