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 M\"obius 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 R\'edei 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.