Thread: 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

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.

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

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 be a finite field with elements.
Define , we know that for all .
Therefore, always has no zero.