As per the title. I'm wondering if $f(x) \in \mathbb{Z}[x]$ is monic and has degree $\geq 2$ and $p$ is prime. I want to prove that if $f \mod n$ in $\mathbb{F}_p$ is irreducible, then $f$ is irreducible in $\mathbb{Q}[x]$. One approach I considered is that since irreducibility over rationals implies irreducibility over ring of integers. So, if I can show that if it's reducible in the integers, it is in $\mathbb{F}_p$ would be an equivalent statement. Could someone show me how I could prove this?

  • $\begingroup$ Shouldn't that be "$\;f\pmod p\;$ in $\;\Bbb F_p\;$ ..." ? $\endgroup$ – DonAntonio Jan 15 '18 at 13:02
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    $\begingroup$ It's not obvious? If $f=gh$ in $\mathbb Q[X]$ then take coefficients $g$ and $h$ by $\equiv (mod p)$ $\endgroup$ – Michael Rozenberg Jan 15 '18 at 13:03

Suppose $\;f(x)\in\Bbb Z[x]\;$ reducible in $\;\Bbb Q[x]\;$ . Then by Gauss Lemma it is reducible over $\;\Bbb Z[x]\;$, say $\;f(x)=g(x)h(x)\;,\;\;g,h\in\Bbb Z[x]\;$ , but then

$$\;f(x)\pmod p=\left(g(x)\pmod p\right)\left( h(x)\pmod p\right)\implies f(x)\pmod p\in\Bbb F_p[x]\,$$

is reducible.


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