Is the polynomial $x^2-xy^5-y^2-1$ an irreducible polynomial?

Question

Fix a field $k$ of characteristic $0$, and let $f=x^2-xy^5-y^2-1\in k[x,y]$. I've succeeded in proving that $f$ is nonsingular at each point, but I'm struggling to prove irreducibility. Is there any useful criterion here?

Answer 1

By Gauss's Lemma, it suffices to prove that $f$ is irreducible in $k(y)[x]$. As you indicate in the comments, since $f$ has degree $2$ in $x$, then $f$ is reducible iff it has a root, which by the quadratic formula (here we need that the characteristic of $k$ is not $2$) occurs iff the discriminant $g := y^{10} + 4y^2 + 4$ is a square. One can show that $\gcd(g, g') = 1$ using the Euclidean algorithm, which implies that $g$ is squarefree, hence is not a square.