TITLE: Irreducible of $\mathbb Z/p\mathbb Z$.
QUESTION [1 upvotes]: What are the irreducible number of $\mathbb Z/p\mathbb Z$ ? It looks strange since in a field it looks complicate to talk about irreducible since all element are invertible. So if my question has no sense, how can I prove using Eisenstein that $X^5+2X+2$ is irreducible over $\mathbb Z/3\mathbb Z$ ?

REPLY [1 votes]: Trying to explain why Eisenstein's criterion cannot be used.
In a field, such as $\Bbb{Z}/p\Bbb{Z}$, all the non-zero elements are units. Therefore a field has no irreducible elements in the sense needed for Eisenstein's criterion.
Eisenstein's criterion can be used to conclude that a polynomial in $K[x]$, $K$ a field, is irreducible only when $K$ is the field of fractions of a suitable integral domain $R$. By suitable I mean an integral domain, where we can prove Gauss's lemma, e.g. a GCD-domain (see the latter Wikipedia article for a description).
The simplest example is the familiar case, when $K=\Bbb{Q}$ and $R=\Bbb{Z}$. Other examples where Eisenstein's criterion can be used to good effect are:

$K$ is the field of $p$-adic numbers $\Bbb{Q}_p$ and $R$ is the ring of $p$-adic integers (denoted $\Bbb{Z}_p$, but should not be confused with the residue class ring of integers $\Bbb{Z}/p\Bbb{Z}$ which is often denoted the same way). Here Eisenstein's criterion is superceded by the use of Newton's polygon, which is more general.
$R$ is the polynomial ring $L[t]$ over some field $L$, and $K=L(t)$ is the corresponding field of fractions, i.e. the field of rational functions over $L$. For example the linear polynomial $t$ is clearly irreducible in $R=\Bbb{Q}[t]$. Therefore over the field $K=\Bbb{Q}(t)$ Eisenstein allows us to conclude that polynomials of the form $x^n-t$ are irreducible in $K[x]$. The irreducible polynomial $t$ takes the role of the prime $p$.
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