For any prime power q, a polynomial \(f(X)\in\mathbb{F}_q[X]\) is “exceptional” if it induces bijections of \(\mathbb{F}_qk\) for infinitely many k; this condition is known to be equivalent to f(X) inducing a bijection of \(\mathbb{F}_qk\) for at least one k with qk ⩾ deg(f)4. In this paper, we introduce the notion of an “exceptional” extension of local fields of any characteristic, and show that if \(f(X)\in\mathbb{F}_q[X]\) is exceptional in the classical sense, then the field extension \(\mathbb{F}_q(X)/\mathbb{F}_q(f(X))\) yields an exceptional local field extension upon passing to the completion at a degree-1 place. We describe all exceptional local field extensions of degree coprime to the residue characteristic, determine the relationship between the exceptionality of a local field extension and the exceptionality of a subextension, and give various Galois-theoretic characterizations of exceptional local field extensions. As a consequence, we obtain three new proofs, using quite different tools, of a theorem of Guralnick and Müller (1997) about ramification indices in exceptional maps between curves over \(\mathbb{F}_q\) . This theorem generalizes a result of Lenstra, which subsumes earlier conjectures of Carlitz and Wan.
