# [FOM] A question on fields

John Baldwin jbaldwin at uic.edu
Mon Aug 5 20:25:00 EDT 2013

The question you ask directly may require a little work.  To show ACF_0 (or
acf for that matter) is not finitely axiomatizable note that any finite set
of equations have a solution in a finite field.

John T. Baldwin
Professor Emeritus
Department of Mathematics, Statistics,
and Computer Science M/C 249
jbaldwin at uic.edu
851 S. Morgan
Chicago IL
60607

On Sun, Aug 4, 2013 at 11:11 AM, SHASHI SRIVASTAVA <smohan53 at gmail.com>wrote:

>  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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