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
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 $\displaystyle F$ 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.