Show $Z(yf-1)$ is irreducible.

Question

Question:

$k$ is an algebraically closed field. Let $f \in k[x_1, \ldots, x_n]$ be an irreducible polynomial. Show that $Z(yf-1)\subseteq \textbf{A}^{n+1}$, with coordinates $x_1, \ldots, x_n, y$, is irreducible.

Attempt:

I tried to use the general approach that the set is irreducible iff $(yf-1)$ is a prime ideal iff the coordinate ring is an integral domain, but there has been no concrete progress. I think the main difficulty is I don't know where to use the condition $f$ is irreducible.

Thanks for help.

Answer 1

Hint. $R[y]/(ay-1)\simeq R[a^{-1}]$. (For more details see Localization in a ring.)