Given a polynomial in more than one variable, how do you prove it's irreducible? I only have one method, called Eisenstein's criterion.
I can't work out the following problem:
Given that $u$ and $v$ are relative prime numbers, and that $f(x_1,x_2,\dots,x_n)$ is an irreducible polynomial over $\mathbb{C}$, prove that $y^u+f^v$ is irreducible (as a polynomial in variables $x_1, \dots, x_n, y$).
If $v=1$, it's trivial. But I think it's hard if $v>1$.
By Kummer theory, the extension $$\mathbb C(x_1,\ldots,x_n) \subseteq \mathbb C(x_1,\ldots,x_n)\left(\sqrt[u]{-f^v}\right)$$ is cyclic of degree $u$, since $u$ is the smallest number $k$ such that $(\sqrt[u]{-f^v})^k \in \mathbb C(x_1,\ldots,x_n)$ (here you need that $u$ and $v$ are coprime, that $\mathbb C[x_1,\ldots,x_n]$ is a UFD, and that $f$ is irreducible).
Thus, the polynomial $y^u + f^v$ is irreducible over $\mathbb C(x_1,\ldots,x_n)$, since it defines a field extension whose degree equals the degree of the polynomial. Since it is monic, Gauß's lemma implies that it is irreducible over $\mathbb C[x_1,\ldots,x_n]$. $\square$