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In transcendental number theory, the Lindemann–Weierstrass theorem is a result that is very useful in establishing the transcendence of numbers. It states that if are algebraic numbers which are linearly independent over the rational numbers then are algebraically independent over in other words the extension field has transcendence degree over An equivalent formulation , is the following: If are distinct algebraic numbers, then the exponentials are linearly independent over the algebraic numbers. This equivalence transforms a linear relation over the algebraic numbers into an algebraic relation over 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 is transcendental for every non-zero algebraic number thereby establishing that 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. ==Naming convention== 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 exponents are required to be rational integers and linear independence is only assured over the rational integers,〔''Sur la fonction exponentielle'', Comptes Rendus Acad. Sci. Paris, 77, pages 18–24, 1873.〕 a result sometimes referred to as Hermite's theorem.〔A.O.Gelfond, ''Transcendental and Algebraic Numbers'', translated by Leo F. Boron, Dover Publications, 1960.〕 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.〔''Über die Ludolph'sche Zahl'', Sitzungsber. Königl. Preuss. Akad. Wissensch. zu Berlin, 2, pages 679–682, 1882.〕 Shortly afterwards Weierstrass obtained the full result,〔''Zu Hrn. Lindemanns Abhandlung: 'Über die Ludolph'sche Zahl' '', Sitzungber. Königl. Preuss. Akad. Wissensch. zu Berlin, 2, pages 1067–1086, 1885〕 and further simplifications have been made by several mathematicians, most notably by David Hilbert. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Lindemann–Weierstrass theorem」の詳細全文を読む スポンサード リンク
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