# Cutting and pasting in Galois theory

I want to ask who was the first to use cut-paste construction in Galois theory.

This question is motivated from the trend in contemporary Galois theory to use patching methods to construct Galois extensions with a given group. For example, Harbater used formal patching to solve the inverse Galois problem over $\mathbb{Q}_p(x)$. Those patching methods are in analogy with the `classical' cutting and pasting constructions of differential geometry. In particular, the proof of Riemann Existence Theorem (saying that there are covers of Riemann surfaces with given groups and ramification) invokes such construction.

Was Riemann the first?

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