# Proving a field is not algebraically closed

• Apr 12th 2009, 02:17 AM
Zinners
Proving a field is not algebraically closed
Hi,

My problem is this: Prove that if p is a prime, then the field Zp is not algebraically closed. I know that using Fermat's little theorem will help but I can't see how it's not closed. Can anybody help please?

Thanks
• Apr 12th 2009, 04:46 AM
berlioz
Using Fermat's little theorem we have x^(p-1)-1=0for any x doesn't equal zero,so the polynomial x^(p-1)-2(suppose p>2)doesn't have the zero points inZp

if p=2,we can find x^2+x+1 hasn't the zero points in Z2.

So field Zp is not algebraically closed.

Quote:

Originally Posted by Zinners
Hi,

My problem is this: Prove that if p is a prime, then the field Zp is not algebraically closed. I know that using Fermat's little theorem will help but I can't see how it's not closed. Can anybody help please?

Thanks

• Apr 13th 2009, 10:10 AM
ThePerfectHacker
Quote:

Originally Posted by Zinners
Hi,

My problem is this: Prove that if p is a prime, then the field Zp is not algebraically closed. I know that using Fermat's little theorem will help but I can't see how it's not closed. Can anybody help please?

Thanks

In general let $F$ be a finite field with $q$ elements.
Define $f(x) = x^q - x +1$, we know that $a^{q-1}=1 \implies a^q - a = 0$ for all $a\in F^{\times}$.
Therefore, $f(x) = x^q - x + 1$ always has no zero.