Irreducibility of a polynomial in two variables

Question

Anyone knows how to verify that the polynomial $$(ax)^n+a_{n-1}x^{n-1}+\cdots +a_1x+a_0-y^n$$ is irreducible, where $n\geq 2$, $a,a_i\in\mathbb{Z}$, $a\neq 0$, and $(ax)^n+a_{n-1}x^{n-1}+\cdots +a_1x+a_0$ is not a $d$-th power in $\mathbb{Z}[x]$ for any divisor $d\geq 2$ of $n$.

Answer 1

Denote the polynomial by $F$, then $F=-(y^n-b)$ where $b\in \mathbb{Q}(x)$.

Verify that the $b$ is satisified the conditions in Theorem 9.1, chapter VI, Part two, in Serge Lang's Algebra, GTM 211, 3rd ed.

Suppose the polynomial, $F$, is not irreducible in $\mathbb{Z}$[x,y], then $-F=GH$ for some non-units $G,H\in \mathbb{Z}[x,y]$. Comparing the coefficient of $y^n$, $G$ and $H$ must be non-constant in $y$, i.e., $G,H$ are of degree $\geq 1$ with respcet to $y$. Thus we have a non-trivial factorizaiton of $-F=y^n-b\in \mathbb{Q}(x)[y]$. So by Theorem 9.1, in chapter VI, Part two in Serge Lang's Algebra, GTM 211, 3rd ed, we obtain a contradiction.