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The Ruled Residue Theorem asserts that given a ruled extension $(K|k,v)$ ofvalued fields, the residue field extension is also ruled. In this paper weanalyse the failure of this theorem when we set $K$ to be algebraic functionfields of certain curves of prime degree $p$, provided $p$ is coprime to theresidue 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 forthe residue field extension to be non-ruled which are formulated only in termsof the values of the coefficients. This provides a far-reaching generalizationof a certain important result regarding non-ruled extensions for functionfields of smooth projective conics.

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