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

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- Apr 12th 2009, 02:17 AMZinnersProving 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 AMberlioz
- Apr 13th 2009, 10:10 AMThePerfectHacker
In general let $\displaystyle F$ be a finite field with $\displaystyle q$ elements.

Define $\displaystyle f(x) = x^q - x +1$, we know that $\displaystyle a^{q-1}=1 \implies a^q - a = 0$ for all $\displaystyle a\in F^{\times}$.

Therefore, $\displaystyle f(x) = x^q - x + 1$ always has no zero.