If polynomials are almost surjective over a field, is the field algebraically closed?

Let K be a field. Say that polynomials are almost surjective over K if for any nonconstant polynomial f(x)K[x], the image of the map f:KK contains all but finitely many points of K. That is, for all but finitely many aK, f(x)a has a root.

Clearly polynomials are almost surjective over any finite field, or over any algebraically closed field. My question is whether the converse holds. That is:

If K is an infinite field and polynomials are almost surjective over K, must K be algebraically closed?

(This answer to a similar question gave a simple proof that C is algebraically closed from the fact that polynomials are almost surjective over C. However, this proof made heavy use of special properties of C such as its topology, so it does not generalize to arbitrary fields.)

Answer

Attribution
Source : Link , Question Author : Eric Wofsey , Answer Author : Community

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