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On reading about the construction of a Riemann surface for algebraic functions, I am having difficulties visualizing why the construction produces a manifold over points that are multiple roots of the defining polynomial. I would be grateful for some clarification.

The example I have in mind is $\sqrt{z}$. I understand how the Riemann surface is a $2$-sheeted cover over nonzero points, construction is completed by adding a single point over $0$. I do not see why the result is a manifold around this added point.

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You need to realize that the Riemann surface of $\sqrt{z}$ is $\mathbb{C}$. Thus there is nothing at all singular about then nbd of $0$. The covering map is $w\mapsto w^2$. There is a theorem that under change of coordinates any analytic map is locally equivalent to $z\mapsto z^n$ where $n$ is the number of sheets. So at an isolated multiple root there is an isomorphism of this deleted nbd with the disc minus the origin and by filling in the point you are using the $z^n$ map to extend the covering map.

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