Exceptional extensions of local fields and the Carlitz--Wan conjecture

For any prime power $q$, a polynomial $f(X)\in {\text{F}}_q[X]$ is exceptional'' if it induces bijections of $\F_{q^k}$ for infinitely many $k$; this condition is known to be equivalent to $f(X)$ inducing a bijection of $\F_{q^k}$ for at least one $k$ with $q^k\ge °(f)^4$. In this paper, we introduce the notion of an exceptional’’ extension of local fields of any characteristic, and show that if $f(X)\in {\text{F}}_q[X]$ is exceptional in the classical sense then the field extension ${\text{F}}_q(X)\mathrm{/}{\text{F}}_q(f(X))$ 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 exceptionality of a local field extension and 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 about ramification indices in exceptional maps between curves over ${\text{F}}_q$. This theorem generalizes a result of Lenstra which subsumes earlier conjectures of Carlitz and Wan.