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  "body": "\n## Statement\n\nLet $\\pi^s_* = \\bigoplus_{n \\ge 0} \\pi^s_n$ be the graded ring of [[stable homotopy groups of spheres|HGZAYPQBXW]], with product given by composition.\nThen every element $\\alpha \\in \\pi^s_n$ with $n \u003e 0$ is nilpotent: $\\alpha^m = 0$ for some $m \\ge 1$.\n\n## Remarks\n\nNishida proved the theorem in 1973.\nHis proof uses extended power constructions on spectra and the operations they induce.\nFor example, the stable Hopf map $\\eta \\in \\pi^s_1$ satisfies $\\eta^4 = 0$, while $\\eta^3 \\ne 0$.\nThe hypothesis $n \u003e 0$ is necessary: $\\pi^s_0 \\cong \\Z$ is generated by the identity, and no nonzero integer is nilpotent.\nThe nilpotence theorem of Devinatz, Hopkins and Smith extends this result from the sphere spectrum to all ring spectra.\n\n## Explanations\n\n### L1 human\n\nStable maps between spheres can be composed, and composing a map of positive degree with itself raises the degree.\nThe theorem says that if one keeps composing such a map with itself, the result is eventually null-homotopic.\nSo no element of positive degree can be iterated forever.\n\n### L2 human\n\nThe ring $\\pi^s_*$ is graded-commutative, and its positive part consists of torsion elements.\nIn a general graded-commutative ring, torsion elements need not be nilpotent, so the theorem is a real constraint on the stable stems.\nIt was the starting point for the chromatic picture of stable homotopy theory, where the nilpotence theorem of Devinatz, Hopkins and Smith characterises nilpotent self-maps through complex cobordism.\n",
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