Show that the polynomial $X^4-10X+1$ is irreducible in $\mathbb{Z}[X]$ but reducible in $\mathbb{F}_p[X]$ for all prime $p$.
I could show the irreducibility in $\mathbb{Z}[X]$ but not sure how to proceed in case of $\mathbb{F}_p[X]$
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How does it factor for $p=17$? – almagest Apr 14 '16 at 10:17
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There is a related question about $x^4+1$: http://math.stackexchange.com/questions/77155/reducible-polynomial-modulo-every-prime?rq=1 – student forever Apr 14 '16 at 10:18
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1This may help for the explicit factorization:http://www.ces.clemson.edu/~jimlb/Teaching/Math581/Math581homework2solutions.pdf – Sarvesh Ravichandran Iyer Apr 14 '16 at 10:20
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Yeah I know how to do it for $X^4+1$ and tried it in the same way. – Rohan Apr 14 '16 at 10:21
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Another one is: http://math.stackexchange.com/questions/160847/polynomials-irreducible-over-mathbbq-but-reducible-over-mathbbf-p-for?rq=1 – student forever Apr 14 '16 at 10:21
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3All fine, but as stated the polynomial is NOT reducible for $p=17$. I assume you meant $X^4-10X^2+1$. – almagest Apr 14 '16 at 10:27
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If you know a little number theory (like Cebotarev's theorem), this amounts to showing that the Galois group of your polynomial over $\mathbb{Q}$ does not contain any element of order $4$.
Captain Lama
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