Exceptional rational functions of degree 5 over finite fields: classification by monodromy, ramification, and the Riemann-Hurwitz formula

We complete the classification of exceptional rational functions of degree 5 over any finite field ${\mathbb{F}}_q$ of characteristic different from 2 and 5, up to Möbius equivalence. Our approach employs the arithmetic and geometric monodromy groups, combined with the ramification structure of the associated cover and the Riemann-Hurwitz formula. The cyclic monodromy case yields precisely monomials and Rédei functions, while the dihedral case is resolved by analysing its inertia groups and branch points; this leads to Dickson polynomials and a non-polynomial family arising from rational 5-isogenies of elliptic curves. We also obtain partial classifications in characteristics 2 and 5: all cases with cyclic geometric monodromy and all dihedral cases admitting an ${\mathbb{F}}_q$-rational branch point are determined. The only unresolved cases are those with geometric monodromy group $D_5$ and no ${\mathbb{F}}_q$-rational branch point.