Does the group $\mu_n$ of $n^{\mathrm{th}}$-roots of unity form an algebraic group over $\mathbb{R}$ for every $n$? If so, why?
I understand that $\mu_n$ forms an algebraic group over $\mathbb{C}$, because it is given by the scheme $\mathrm{Spec}\, \mathbb{C}[x]/(x^n-1)$, but I don't see how to view it as an algebraic group over $\mathbb{R}$.
Perhaps this is trivial, because every scheme over $\mathbb{C}$ can be viewed as a scheme over $\mathbb{R}$, but I might be missing something here.