Rational separability of the integral closure

We investigate the following question. Let K be a global field, i.e. a number field or an algebraic function field of one variable over a finite field of constants. Let WK be a set of primes of K, possibly infinite, such that in some fixed finite separable extension L of K, all the primes of WK do not have factors of relative degree 1. Let M be a finite extension of K and let WM be the set of all the M-primes above the primes of WK. Then does WM have the same property? The answer is “always” for one variable algebraic function fields over finite fields of constants and “not always” for number fields. In this paper we give a complete description of the conditions under which WM inherits and does not inherit the above described property.