1

Give $f(x)=a_0+a_1x+\ldots+a_nx^n$ and prime $p$ that $p \nmid a_n$ and $GCD(a_1,a_2,\ldots,a_n)=1$. Which one in two clause below is correct?

(1):"If $f(x)$ is irreducible in $\mathbb{Z}[x]$ then $f(x)$ is irreducible in $\mathbb{Z}_p[x]$.

(2):"If $f(x)$ is irreducible in $\mathbb{Z}_p[x]$ then $f(x)$ is irreducible in $\mathbb{Z}[x]$".

I have prove that (1) is wrong and i think (2) is correct but i can't prove it.

Xiuwei Lee
  • 121
  • 2
    If we denote by $\overline{f}$ the reduction of a polynomial $f$ modulo $p$, then you should be able to verify that for all polynomials in $\Bbb{Z}[x]$ we have the implication $$f(x)=g(x)h(x)\implies \overline{f}(x)=\overline{g}(x)\overline{h}(x).$$ The claim follows from this together with Gauss lemma and the assumption $p\nmid a_n$. Do you see why? – Jyrki Lahtonen May 31 '20 at 04:43
  • 2
    This has been handled (not necessaily very verbosely) many times on our site. See 1, 2 linking to here. It gives a useful way of proving the irreducibility of polynomials in $\Bbb{Z}[x]$. See there for a typical deduction. – Jyrki Lahtonen May 31 '20 at 04:54

0 Answers0