Prove or disprove.
Let $f_n(x)=x^{n-1}+x^{n-2}+\cdots +x+1$. Then $f_p(x^{p^{e-1}})$ is irreducible in $\mathbb Q[x]$ for all prime $p$.
I know if $p$ is a prime $f_p(x)$ is $p$-th cyclotomic polynomial and is irreducible.
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1$f_p(x^{p^{e-1}})$ is the $p^e$th cyclotomic polynomial and hence irreducible. – Jyrki Lahtonen Sep 13 '19 at 16:48
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How to prove that fact – sabeelmsk Sep 13 '19 at 16:52
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The most general case is covered here. In this special case this approach should work even though it is nominally only about $e=2$. – Jyrki Lahtonen Sep 13 '19 at 16:59
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To become $p^e$-th cyclotomic polynomial, degrees are not matching.$\phi(p^e)\ne p^{e-1}$. – sabeelmsk Sep 13 '19 at 23:32
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I got the method of answer from the quoted question and answer. – sabeelmsk Sep 14 '19 at 02:56
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But is this polynomial realy cyclotomic polynomial? – sabeelmsk Sep 14 '19 at 02:57