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).

REFERENCES:

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. Paris 77, 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.