# Polynomial $x^5 + 5x^2 +1$ is irreducible or not

Which way I can determine whether the polynomial $x^5 + 5x^2 +1$ is irreducible over $\mathbb Q$ or not?

Mod $p$ Irreducibility Test and Eisenstein's criterion not applicable here.

Which way I should proceed?

• You mean you cant use the mod P irreducibility or that you tried without success? – AHandsomeAlien Jun 29 '17 at 4:29
• Reduce modulo $2$. Show that your new polymomial has no roots in $\Bbb Z_2$ and show that your reduced polynomial can't be written as $q(x)p(x)$ where $q$ has degree $2$ and $p$ has degree $3$. – Rocket Man Jun 29 '17 at 4:31
• No. Actually I tried. But then after reducing by mod5 polynomial becomes x^5 + 1 . Then I dont know how should I check if x^5 +1 is irreducible in Z5 or not. – user458361 Jun 29 '17 at 4:32
• I second AJ Stas's advice. Reduce modulo 2. The polynomial $x^5+x^2+1$ has no zeros in $\Bbb{Z}_2$, and it is not divisible by the unique (!) quadratic irreducible either. – Jyrki Lahtonen Jun 29 '17 at 4:41
• By the way, $x^5 + 1$ is not irreducible mod $5$ because $-1$ is a root. – cat Jun 29 '17 at 7:29

Try $x=y-1$. It should help by the Eisenstein's criterion.
$7^5+5\cdot 7^2+1 = 17053$ which is a prime number. Thus the polynomial is irreducible by Cohn's irreducibility criterion.