# Unibranch local ring

Theorem (EGA, III.4.3.7) Let X and Y be two integral locally noetherian schemes and ${\displaystyle f\colon X\to Y}$ a proper dominant morphism. Denote their function fields by K(X) and K(Y), respectively. Suppose that the algebraic closure of K(Y) in K(X) has separable degree n and that ${\displaystyle y\in Y}$ is unibranch. Then the fiber ${\displaystyle f^{-1}(y)}$ has at most n connected components. In particular, if f is birational, then the fibers of unibranch points are connected.