A note on trascendental elements of an algebraic function field of one variable

Let K be an algebraic function field of one variable with a constant field k. We shall assume that K is separably generated over k and then we shall denote by x a separating element of K over k. The purpose of this note is to discuss the problem whether we can find the elements a, b, c, d, in k satisfying ad-bc≠O such that every prime divisor of K that divides the denominator divisor or the numerator divisor of the elemerit of the from ax+b/cx+d is unramified over the rational function field k(x), or not. It is obvious that we can find such elements if k is not finite. But it is impossible in general if k is finite. In §2, we shall prove that there exist such elements in k under some condition, and we shall show that this is the best condition in a sense in §3