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I had an exercise were I had to prove that for each natural $n$ and each prime $p$, there exists an irreducible polynomial in $\mathbb{Z}_p[X]$. But then I was searching for an example were for a certain $n$ (not prime), there are no irreducible polynomial of degree $n$. So I considered the easiest example $\mathbb{Z}_4[X]$ but :

degree 0 : $2$ is irreducible

degree 1 : $X$ is irreducible

degree 2 : $X^2+1$ is irreducible

is there a $n$ for which we don't have irreducible polynomial?

roi_saumon
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  • Why do people use $\mathbb{Z}_p$ in algebra/number theory when they don't mean the $p$-adic integers? – user10354138 Sep 28 '18 at 21:07
  • I think that if $f(x) \in \mathbb{Z}[x]$ is an irreducible monic polynomial modulo $p$, and if $p$ divides $n$, then $f(x)$ is also irreducible modulo $n$. (Proof: if it factored modulo $n$, it would also factor modulo $p$. The reason I said monic was to try to ensure that none of the factors could be $1$ modulo $p$.) So I don't think any example exists for what you requested. Do you believe this reasoning? – CJD Sep 28 '18 at 21:11
  • See https://math.stackexchange.com/questions/2539493/defining-irreducible-polynomials-over-polynomial-rings/2539517#2539517 and https://math.stackexchange.com/a/2906963/589 – lhf Sep 28 '18 at 22:29

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