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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?

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

1 Answer 1

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

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

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  • $\begingroup$ 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

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