In book THE THEORY OF ALGEBRAIC NUMBERS by HARRY POLLARD page 34:
Let p be a prime and consider the so-called cyclotomic polynomial
$\frac{x^{p}-1}{x-1} = x^{p-1}+x^{p-2}+…+x+1$.
This is irreducible over Q if
$\frac{(x+1)^{p}-1}{(x+1)-1} =\frac{(x+1)^{p}-1}{x}$
is also.But the latter is of the form (why?)
$x^{p-1}+p(x^{p-2}+…)+p$,
and the irreducibility follows directly from Theorem 3.8(Eisenstein’s irreducibility criterion)
As another important example consider the polynomial
$\frac{x^{p^2}-1}{x^p-1} = x^{p(p-1)}+x^{p(p-2)}+…+x^p+1$.
Replace $x$ by $x+1$ yields
$x^{p(p-1)}+p*q(x)$,
where $q(x)$ has integral coefficients and final term $1$. Once again Eisenstein’s criterion shows that the polynomial is irreducible over $\mathbb{Q}$.
My problem is how to know that $q(x)$ has integral coefficients and its final term is $1$.