Irreducibility of $f_p(x^{p^{c-1}})$ when given $f_n(x)=x^{n-1} +x^{n-2}+.....+x+1$ where $n$ is positive integer.

Question

For a positive integer $n$ let $f_n(x)=x^{n-1} +x^{n-2}+.....+x+1$ then what can we say about $\displaystyle f_p(x^{p^{c-1}})$ For every prime $p$ and every positive integer $c$ , is it reducible or irreducible over $\mathbb{Q}[x]$.

Solution i tried - I know that for $f_p(x)=x^{p-1}+x^{p-2}+....+x+1 $ is a cyclotomic polynomial and irreducible over $\mathbb{Q}[x]$ but how can i use this to show that $\displaystyle f_p(x^{p^{c-1}})$ is reducible or irreducible over $\mathbb{Q}[x]$ ?

please help

Thank you

Answer 1

$f_p(x^{p^{c-1}})$ is the cyclotomic polynomial for $p^c$. See here: https://en.wikipedia.org/wiki/Cyclotomic_polynomial

Also, this proof works. Irreducibility of prime degree cyclotomic polynomials