Let
and
be algebraic numbers such that the s differ from zero and the s differ from each other. Then the expression
cannot equal zero. The theorem was proved by Hermite (1873) in the special case of the s
and s
rational integers, and subsequently proved for
algebraic numbers by Lindemann in 1882 (Lindemann 1888). The proof was subsequently
simplified by Weierstrass (1885) and Gordan (1893).
Dörrie, H. "The Hermite-Lindemann Transcendence Theorem." §26 in 100
Great Problems of Elementary Mathematics: Their History and Solutions. New
York: Dover, pp. 128-137, 1965.Hermite, C. "Sur la fonction
exponentielle." Comptes Rendus Acad. Sci. Paris77, 18-24, 1873.Gordan,
P. "Transcendenz von und ." Math. Ann.43, 222-224, 1893.Lindemann,
F. "Über die Ludolph'sche Zahl." Sitzungber. Königl. Preuss.
Akad. Wissensch. zu Berlin No. 2, pp. 679-682, 1888.Weber,
H. Lehrbuch
der Algebra, Vols. I-II. New York: Chelsea, 1902.Weierstrass,
K. "Zu Hrn. Lindemann's Abhandlung: 'Über die Ludolph'sche Zahl.' "
Sitzungber. Königl. Preuss. Akad. Wissensch. zu Berlin No. 2, pp. 1067-1086,
1885.