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

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
• Wouldn't I be double counting? – user261688 Aug 13 '15 at 0:34
• I cannot understand the third link and the rest uses things I haven't proven yet. Could you explain what you meant by counting? – user261688 Aug 13 '15 at 0:42
• 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