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Is the polynomial $2x^{10}+25x^3+10x^2-30$ irreducible over $\mathbb Q$?

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Try this with $p=5$. – awllower Mar 7 '13 at 16:05

Hint: $p(x)\in \mathbb{Z}[x]$ is irreducible $\mathbb{Z}[x]$ implies that it is irreducible in $ \mathbb{Q}[x]$. Now verify that $2x^{10}+25x^3+10x^2-30\,$ is irreducible in $\mathbb{Z}[x]$. Do this by using Eisenstein's criterion with $p=5$

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thank you for your help. – user65562 Mar 7 '13 at 16:16

This polynomial is primitive so you can use Eisenstein to verify if it is irreducible. Take the prime $5$ and notice that $5$ doesn't divides $2$, $5^2$ doesn't divides $30$, but $5$ divides $25$ and $10$, therefore, this polynomial is irredducible over $\mathbb{Z}$ and over $\mathbb{Q}$.

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you mean "doesn't" – Martin Brandenburg Mar 8 '13 at 0:19
Problem solved. Thanks! ^^ – Integral Mar 8 '13 at 1:17

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