Irreducibility of $p^{n-1}X^n+pX+1$ over $\mathbb{Q}$ [duplicate]
I am attempting to show that $f(X)=p^{n-1}X^n+pX+1$ are irreducible over $\mathbb{Q}$ for any positive integer $n$ and any prime $p$. At the behest of my teacher, and their hint, I would like to do so using Eisenstein's criterion on $p f(X)$, but I am having issues seeing how Eisenstein's is applicable here since $p$ will then be dividing the leading coefficient of $pf(X)$.
Solution 1:
By the comment of lhf we have that $g(X)=X^n+pX+p$ is irreducible by Eisenstein's criterion, and $g(pX)=pf(X)$.