[FOM] A question on fields

[SHASHI SRIVASTAVA]

 How does one prove the following?

For every $d \geq 2$, there is a field $K$ such that every polynomial in
$K[X]$ of degree $\leq d$ has a root in $K$ but $K$ is not algebraically
closed.

This will imply that the theory of algebraically closed fields is not
finitely axiomatizable, which is my main interest.

Shashi M. Srivastava
Indian Statistical Institute
Kolkata
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