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**Dark Sun** Prove that for every prime p, there is an irreducible quadratic in $\displaystyle Z_p[x]$ (the polynomial ring of the integers mod p).

I have not been able to wrap my brain around this one. Does anyone have any ideas?

I have been looking at $\displaystyle x^2+x+1$ and $\displaystyle x^2+1$, but haven't been able to derive a contradiction.

This is what I have so far: Assume $\displaystyle x^2+1$ is reducible in $\displaystyle Z_p[x]$. Then, $\displaystyle \exists a,b\in Z_p[x]$ s.t. $\displaystyle x^2+1=(x-a)(x-b)=x^2-(a+b)x+ab\Rightarrow a+b=0, ab=1$

Thanks!