# Unibranch local ring

In algebraic geometry, a local ring *A* is said to be **unibranch** if the reduced ring *A*_{red} (obtained by quotienting *A* by its nilradical) is an integral domain, and the integral closure *B* of *A*_{red} is also a local ring.Template:Fact A unibranch local ring is said to be **geometrically unibranch** if the residue field of *B* is a purely inseparable extension of the residue field of *A*_{red}. A complex variety *X* is called **topologically unibranch** at a point *x* if for all complements *Y* of closed algebraic subsets of *X* there is a fundamental system of neighborhoods (in the classical topology) of *x* whose intersection with *Y* is connected.

In particular, a normal ring is unibranch. The notions of unibranch and geometrically unibranch points are used in some theorems in algebraic geometry. For example, there is the following result:

**Theorem** Template:Harv Let *X* and *Y* be two integral locally noetherian schemes and 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 is unibranch. Then the fiber has at most *n* connected components. In particular, if *f* is birational, then the fibers of unibranch points are connected.

In EGA, the theorem is obtained as a corollary of Zariski's main theorem.