Is there an **infinite** field $k$ together with a polynomial $f \in k[x]$ such that the associated map $f \colon k \to k$ is not surjective but misses only finitely many elements in $k$ (i.e. only finitely many points $y \in k$ do not lie in the image of $f$)?


For finite fields $k$, there are such polynomials $f$. If such a poynomial $f$ exists, then $k$ cannot be algebraically closed; the field $\mathbb{R}$ doesn't work either.



Can a non-surjective polynomial map from an infinite field to itself miss only finitely many points?
Since such a polynomial would have to have degree at least 2, its existence implies that 
the set of k-rational points of the affine line over k is *thin* in the sense of Serre's *Topics In Galois Theory*. It follows from the results presented in that book that this cannot be the case over any *Hilbertian field*. This includes finite extensions of Q, finite extensions of F(t) for any field F, and many other fields. 


What about p-adic fields?


