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

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 $K= k(X)(\sqrt[p]{aX^p+bX+c})$ where $a\neq 0$. 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.