Give an example of a polynomial $p(x) \in \mathbb{Z}[x]$ of degree 10 which is reducible modulo 2,3 and 5 but irreducible over $\mathbb{Z}$.

Asked Mar 25 '16 at 23:55

Active Mar 25 '16 at 23:55

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I tried to solve this by Eisenstein Criterion,let $p(x)$ is irreducible over $\mathbb{Z}$, but I don't know how to let it reducible module 2,3, and 5. Please help me to figure it out! Thank you!

asked Mar 25 '16 at 23:55

Nhay

1

Try a very simple form...like $x^{10}+2$. That clearly factors $\pmod 2$ and $\pmod 3$ at least. – lulu Mar 26 '16 at 00:01
2

Or, even easier, $x^{10}+30$. But my first example works as well. – lulu Mar 26 '16 at 00:09
Simpler yet, $x^2 + 30$? – Théophile Mar 26 '16 at 01:24
@Théophile The OP specified degree $10$, though of course $x^n+30$ does work for all $n$. – lulu Mar 26 '16 at 01:25
@lulu Oh, I see; thanks. I missed that. – Théophile Mar 26 '16 at 01:25

Give an example of a polynomial $p(x) \in \mathbb{Z}[x]$ of degree 10 which is reducible modulo 2,3 and 5 but irreducible over $\mathbb{Z}$.

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