A ruled residue theorem for algebraic function fields of curves of prime degree

Abstract

The Ruled Residue Theorem asserts that given a ruled extension (K|k,v) of valued fields, the residue field extension is also ruled. In this paper we analyse the failure of this theorem when we set K to be algebraic function fields of certain curves of prime degree p, provided p is coprime to the residue characteristic and k contains a primitive p-th root of unity. Specifically, we consider function fields of the form K=k(X)(aXp+bX+cp) where a≠0. We provide necessary conditions for the residue field extension to be non-ruled which are formulated only in terms of the values of the coefficients. This provides a far-reaching generalization of a certain important result regarding non-ruled extensions for function fields of smooth projective conics.