Show that the polynomial $x^{n-1}+x^{n-2}+\cdots+x+1$ is irreducible over $\Bbb Q$ if and only if $n=p$ is a prime. (For the direction when $n=p$, make a change of variable $x\to x+1$ and use Eisenstein criterion).
Extra Credit: If $f$ is an irreducible integral polynomial in $\Bbb Q[x]$, can $f$ be changed, with a change of variable, to something satisfying Eisenstein criterion?
My idea:
For the first part, I tried to prove by contradiction. I assume if there is $g(x)$ whose degree is less than $n-1$ and $g(x)\mid f(x)$ there must be another polynomial $h(x)$ whose degree is less than $g(x)$ and $h(x)\mid f(x)$. but I still have no idea how to write a complete prove, and can anyone give me a hint for extra credit part?
Thank you all for reading this.