Generalized primitive elements for transcendental field extensions

Abstract

Let L be a finitely generated separable extension of a field K of characteristic p^ 0. Artin's theorem of a primitive element states that if L is algebraic over K, then L is a simple extension of K. If L is non-algebraic over K, then an element G L with the property L = L'() for every L', LD L'D K, such that L is separable algebraic over L' is called a generalized primitive element for L over K. The main result states that if [K: K p ] > p, then there exists a generalized primitive element for L over K. An example is given showing that if [K : K p ] ^ p, then L need not have a generalized primitive element over K.