How to I use the Eisenstein Criterion to show that for any prime $p$, $1+x^{p}+x^{2p}+...+x^{p(p-1)}$ is irreducible in $\mathbb{Z}[x]$? I tried using the substitution $x = y+1$ and I guess this works but this method seems too complicated. Is there a more efficient/elegant way?
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We only need Eisenstein to show the irreducibility for the cyclotomic polynomial $\Phi_p=1+X+X^2+\cdots +X^{p-1}$ over $\Bbb Q$. For prime powers then the usual proof works for showing that $\Phi_n$ is irreducible by considering $X^n-1=g\Phi_n$ and passing to $\Bbb F_p[X]$. In our case $$ \Phi_{p^2}=1+X+X^p+\cdots +X^{(p-1)p}. $$
Dietrich Burde
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How is the last equation related to Phi_n? – Debbie Mar 11 '20 at 15:24
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Can you also elaborate when you say passing to F_p[x]? I am having trouble seeing the connection between the last two sentences – Debbie Mar 11 '20 at 15:25
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@Dietrich What happened to you dupe searching? Obviously this is a FAQ (and an easy search). – Bill Dubuque Mar 11 '20 at 15:25
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@Dietrich Did you find an answer by me? I didn't answer in the dupe(s) I linked. – Bill Dubuque Mar 11 '20 at 15:31
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I got lost in too many posts about this topic, e.g., here and the links, I am sorry. But I have seen an answer of yours for the general case. – Dietrich Burde Mar 11 '20 at 15:36
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@Dietrich and BillDubuque Thank you very much. Appreciate it! – Debbie Mar 11 '20 at 15:38