In algebraic topology, the nilpotence theorem gives a condition for an element in the homotopy groups of a ring spectrum to be nilpotent, in terms of the complex cobordism spectrum ${\displaystyle \mathrm {MU} }$. More precisely, it states that for any ring spectrum ${\textstyle R}$, the kernel of the map ${\textstyle \pi _{\ast }R\to \mathrm {MU} _{\ast }(R)}$ consists of nilpotent elements.^[1] It was conjectured by Douglas Ravenel (1984) and proved by Ethan S. Devinatz, Michael J. Hopkins, and Jeffrey H. Smith (1988).

Goro Nishida (1973) showed that elements of positive degree of the homotopy groups of spheres are nilpotent. This is a special case of the nilpotence theorem.

• Ravenel's conjectures