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We calculate higher Hochschild homology ${\\sf HH}_*^{[n]}(k[x])$ over $k$ for any integral domain $k$, and ${\\sf HH}_*^{[n]}(\\mathbb{F}_p[x]/x^{p^\\ell})$ for all $n>0$. We use this and \\'etale descent to calculate ${\\sf HH}_*^{[n]}(\\mathbb{F}_p[G])$ for all $n>0$ for any cyclic group $G$, and therefore also for any finitely generated abelian group $G$. 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