ON THE PURITY OF THE BRANCH LOCUS OF ALGEBRAIC FUNCTIONS

1. Let V/k be an absolutely irreducible, r-dimensional normal algebraic variety and let K = k(V) be the function field of V/k; here k denoted an arbitrary ground field. Let K* be a finite separable algebraic extension of K, let k* be the algebraic closure of k in K*, and let V*/k* be a normalization of V in K*. Let P* be an arbitrary point of V* (not necessarily algebraic over k), and let P be the corresponding point of V. We denote by o the local ring of P on V/k and by m the maximal ideal of o. Let o* and m* have a similar meaning for P* and V*/k*. It is well known that: (1) o*m is a primary ideal, with m* as associated prime ideal; (2) the residue field k*(P*) (= o*/m*) is a -finite algebraic extension of the field k(P) (= o/m). Definition: The point P* is said to be unramified (with respect to V) if thefolltwing conditions are satisfied: