Prove that $Y^{n} - X$ is irreducible in $R[X, Y]$ where $R$ is a field, for all $n \in \mathbb{Z}_{\geq 0}$.
I've made an attempt at this problem using the Eisenstein Criterion, would someone be so kind as to check its validity?
Proof Attempt: Note that $R[X, Y] = (R[X])[Y]$. First we show that $Y^{n} - X$ is an Eisenstein polynomial. Because $R$ is a field it is vacuously a UFD, then $R[X, Y]$ is also a UFD. Let $K[X]$ be the field of fractions of $R[X]$
Consider $x \in R[X]$, since $x$ is a linear polynomial its an irreducible element in $R[X]$.
Then clearly, $x \nmid 1$, $x \mid x$ and $x^{2} \nmid x$, as such $Y^{n} - X$ is an Eisenstein polynomial. It follows that $Y^{n} - X$ is irreducible in $K[X, Y]$. Note that $Y^{n} - X$ is a primitive polynomial. Since the polynomial is primitive and irreducible in $K[X, Y]$ it is also irreducible in $R[X, Y]$ $\square$.