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This question talks about fields in which every polynomials are almost surjective, while I am interested in the following case:

$\Bbb F$ is a field such that for every non-constant polynomial $f$ over $\Bbb F$, $f$ or $f-1$ has a root in $\Bbb F$.

If $\Bbb F$ is not the field with two elements, must $\Bbb F$ be algebraically closed?

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  • $\begingroup$ Oh, we should avoid constant polynomial as we do in the definition of algebraically closed field. $\endgroup$ – sawdada Jul 28 '17 at 6:21
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    $\begingroup$ Note that this is equivalent to asking that $f-a$ has a root for all but at most one $a$, since if there are two exceptions you can compose with a linear polynomial to make them $0$ and $1$. $\endgroup$ – Eric Wofsey Jul 28 '17 at 6:26

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