In one of my proof for my assignment I reached a point where I have to prove that $x^9-t^9$ is irreducible in $\mathbb{Z}_7(t^9)[x]$. I am unsure weather this is irreducible. If it is, how do I prove it? Thanks in advance.
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1What is $\mathbb{Z}_7(t^9)$? Is it the field of fractions of $\mathbb{Z}_7[t^9]$ – Alexander Vlasev Oct 21 '12 at 05:09
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1Hint: Eisenstein's Criterion. – Rankeya Oct 21 '12 at 06:00
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@Rankeya: There is no prime element in $Z_7(t^9)$. – user44322 Oct 21 '12 at 21:07
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@AleksVlasev Yes that is the field of fractions. – user44322 Oct 21 '12 at 21:07
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Agreed, but there is a prime element in $\mathbb{Z}_7[t^9]$, which is a ring, not a field. – Rankeya Oct 21 '12 at 21:45