[Chapter 0 (23.2.1), EGA4]

Definition 15.106.1. Let $A$ be a local ring. We say $A$ is unibranch if the reduction $A_{red}$ is a domain and if the integral closure $A'$ of $A_{red}$ in its field of fractions is local. We say $A$ is geometrically unibranch if $A$ is unibranch and moreover the residue field of $A'$ is purely inseparable over the residue field of $A$.

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