Complete Classification of permutation rational functions of degree three over finite fields
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Let q be a prime power, F_q be the finite field of order q and F_q(x) be the field of rational functions over F_q. In this paper we classify all rational functions φ∈ F_q(x) of degree 3 that induce a permutation of P^1( F_q). Our methods are constructive and the classification is explicit: we provide equations for the coefficients of the rational functions using Galois theoretical methods and Chebotarev Density Theorem for global function fields. As a corollary, we obtain that a permutation rational function of degree 3 permutes F_q if and only if it permutes infinitely many of its extension fields. As another corollary, we derive the well-known classification of permutation polynomials of degree 3.
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