Recognizing Simple Subextensions of Purely Transcendental Field Extensions

Abstract.

Let k(X)/k be a finitely generated purely transcendental extension of fields with transcendence basis X. From a generalization of Lüroth's theorem it is known that all intermediate fields K of k(X)/k with (K/k) = 1 are of the form K = k(f) for some f∈k(X). In this note we give a criterion which allows to decide effectively whether the extension K/k is simple if a finite generating set of K over k is known and computations in K are effective. In the affirmative case a primitive element of K over k is computed.