How to show that

$4x^9 -9x^3 + 24x + 13 $ is irreducible over $\mathbb{Q}$.

Since the polynomial is primitive over $\mathbb{Z}$, hence I can show instead that it is irreducible over $\mathbb{Z}$.

I'm thinking of using $\mod p$ irreducibility test here, but it's still hard to prove it since it's in degree 9. Meaning I will have to check individually that it does not have linear, quadratic, etc. factors.

Do you know of any easier way to prove this?

  • $\begingroup$ WA says that the polynomial is irreducible mod $11$. $\endgroup$
    – lhf
    Mar 1, 2018 at 10:19

1 Answer 1


Use $x=y-1$ and Eisenstein for $p=3$.

Since $$-4+9-24+13=-6$$ is not divided by $9$, we are done!

  • $\begingroup$ Ohh i see what you did there. Smart! Thanks! $\endgroup$
    – Jonelle Yu
    Mar 1, 2018 at 7:39
  • $\begingroup$ How did you think of translating it to x-1, and then coming up with an equation where eisenstein's criterion is readily applicable? $\endgroup$
    – Jonelle Yu
    Mar 1, 2018 at 7:50
  • $\begingroup$ Because all $\binom{9}{k}$, where $k\neq0$ and $k\neq9$ they are divided by $3$. Also, coefficients $-9$ and $24$ they are divided by $3$ and I checked that $-4+9-24+13$ is nod divided by $9$ and it turned out. $\endgroup$ Mar 1, 2018 at 8:08

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.