How to show there are irreducible polynomials of any degree over $\Bbb Z _p$? [duplicate]

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How do I show there are irreducible polynomials of any degree in $\mathbb{Z}_p[x]$, with $p$ prime?

I tried counting the number of reducible polynomials of any degree but that turned out to be hard... Any help?

marked as duplicate by egreg, Ken, Michael Galuza, Cameron Buie, Jyrki LahtonenAug 28 '15 at 6:52

• You are on the right track. You can count the monic polynomials of degree $n$ in $\mathbb{Z}_p[X]$ and subtract the ones that have a factor of degree at most $n/2$. What remains is the number of irreducible polynomials there of degree $n$. – hardmath Aug 13 '15 at 0:27
• Do you know the fact that $X^{p^n} - X$ is the product of all irreducible polynomials over $\mathbb{Z}_p$ whose degree divides $n$? This Answer by Marc van Leeuwen leverages that fact to show at least one irreducible polynomial of degree exactly $n$ exists. – hardmath Aug 13 '15 at 1:38