Let $k$ be a field. How do you prove that the algebraic subgroups of $\mathbb{G}_m=\mathrm{Spec}\,k[x, x^{-1}]$ are $\mathbb{G}_m$ itself and the roots of unity $\{x^n=1\}$ for some $n$?
algebraic subgroups of $\mathbb{G}_m$
roots99
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