# [FOM] A question on fields

Gergely Székely gergely.szekely at gmail.com
Mon Aug 5 05:40:06 EDT 2013

Dear Shashi M. Srivastava,

That the theory of algebraically closed fields is not finitely
axiomatizable can be found, for example, in
- Dalen: Logic and Structure (http://books.google.hu/books?id=u0wlXPHATDcC),
see Corollary 4.2.13 on p.109
- Barwise: Handbook of Mathematical Logic (
http://books.google.hu/books?id=b0Fvrw9tBcMC), see Theorem 5.5 on p.119.
- Bell & Slomson: Models and ultraproducts, see Theorem 3.22 on p.101
http://archive.org/stream/ModelsAndUltraproducts/BellSlomson-ModelsAndUltraproducts#page/n109/mode/2up

Best regards,
Gergely Székely

On Sun, Aug 4, 2013 at 6:11 PM, 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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