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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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What is $\mathbb{Z}_7(t^9)$? Is it the field of fractions of $\mathbb{Z}_7[t^9]$ –  Aleks Vlasev Oct 21 '12 at 5:09
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Hint: Eisenstein's Criterion. –  Rankeya Oct 21 '12 at 6:00
    
@Rankeya: There is no prime element in $Z_7(t^9)$. –  user44322 Oct 21 '12 at 21:07
    
@AleksVlasev Yes that is the field of fractions. –  user44322 Oct 21 '12 at 21:07
    
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

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