How to prove a polynomial irreducible over $\mathbb{C}$

Solution 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 $\mathbb C[x_1,\ldots,x_n]$. $\square$