# Question about how to use Eisenstein's criterion

So I have this exercise:

I have to show if $$f:=2X^5-6X+6 \in \mathbb{Z}[X]$$ is irreducible in $$\mathbb{Z}[X]$$.

So Clearly is reducible because $$f$$ can be written as $$f=2(X^5-3X+3)$$. But here comes my question:

If I use Eisenstein's criterion with $$p=3$$ I get that is an irreducible polynomial since $$p$$ divides each $$a_i$$ for $$0 ≤ i < n$$, $$p$$ does not divide $$a_n$$, and $$p^2$$ does not divide $$a_0$$. Where is my error?

• What version of Eisenstein are you using that makes you think it fails here? The most common form is for polynomials over $\Bbb Q[x],\,$ where $2$ is a unit. Commented Dec 15, 2022 at 11:56
• @BillDubuque Eisenstein’s criterion doesn’t make sense for coefficients on a field because there every non zero element divides all the elements of the field, and in addition we don’t have a notion of prime element. Commented Dec 15, 2022 at 12:31
• @Carnby You misunderstood. My point is that EC is often stated in the form of a sufficient test for irreducibility of polynomials over $\Bbb Q[x],\,$ e.g. see the first paragraph of the EC wikipedia page. Commented Dec 15, 2022 at 14:48

• sorry but my definition says: (translated from germany): Let $P(x)$ be a polynomial with integer coefficients, i.e. $P(x)=a_{n}x^{n}+\cdots +a_{1}x+a_{0}$ in $\mathbb {Z}[x]$. If a prime number $p$ exists that divides all coefficients $a_{0}$ through $a_{n-1}$ but $p^2$ does not divide coefficient $a_{0}$ and does not divide $a_{n}$ at all; if $p\mid a_{i}$ for all $i<n$ and $p^{2}\nmid a_{0}$ and $p\nmid a_{n}$ holds, then $P(x)$ is irreducible in $\mathbb{Z}[x]$. Commented Dec 15, 2022 at 12:37
• @MarcoDJ01 That form of the criterion gives irreducibility in $\mathbb{Q}[x]$, which is equivalent to irreducibility in $\mathbb{Z}[x]$ as long as the GCD of the coefficients of the polynomial is 1. So, your polynomial is irreducible in $\mathbb{Q}[x]$ but not in $\mathbb{Z}[x]$; Eisenstein’s criterion seems to fail because your polynomial is not primitive. Commented Dec 15, 2022 at 13:04