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In algebraic topology, the nilpotence theorem gives a condition for an element of the coefficient ring of a ring spectrum to be nilpotent, in terms of complex cobordism. It was conjectured by Ravenel (1984) and proved by Devinatz, Hopkins & Smith (1988).

Nishida's theorem

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.
References

Devinatz, Ethan S.; Hopkins, Michael J.; Smith, Jeffrey H. (1988), "Nilpotence and stable homotopy theory. I", Annals of Mathematics. Second Series 128 (2): 207–241, doi:10.2307/1971440, ISSN 0003-486X, MR 960945
Nishida, Goro (1973), "The nilpotency of elements of the stable homotopy groups of spheres", Journal of the Mathematical Society of Japan 25 (4): 707–732, doi:10.2969/jmsj/02540707, ISSN 0025-5645, MR 0341485.
Ravenel, Douglas C. (1984), "Localization with respect to certain periodic homology theories", American Journal of Mathematics 106 (2): 351–414, doi:10.2307/2374308, ISSN 0002-9327, MR 737778 Open online version.
Ravenel, Douglas C. (1992), Nilpotence and periodicity in stable homotopy theory, Annals of Mathematics Studies, 128, Princeton University Press, ISBN 978-0-691-02572-8, MR 1192553

The electromagnetic field of a plane wave without sources is nilpotent when it is expressed in terms of the algebra of physical space.[6]






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