Some irreducible polynomial

Question

Is the polynomial given by $y^2-p(x)\in C[x,y]$ with $p$, all of whose roots are distinct, an irreducible polynomial? Interesting is when $p$ has degree $3$ innit

Answer 1

Assume you could factor $y^2-p(x)=F(x,y)G(x,y)$ for some polynomials nonconstant $F$ and $G$. Consider the highest order terms of the polynomials $F(x,\cdot)$ and $G(x,\cdot)$ (in the variable $y$). Their product must be $y^2$, so the highest order coefficients must be independent of $x$. Thus $F(x,y)=y+f(x)$ and $G(x,y)=y+g(x)$ for some polynomials $f$ and $g$.

We have $y^2-p(x)=y^2+y(f(x)+g(x))+f(x)g(x)$, so $f=-g$ and $p=f^2$. Therefore $p$ must be a square of another polynomial. In fact, $y^2-p(x)$ is irreducible if and only if $p$ is not a square.

A third order polynomial is surely not a square, so your $y^2-p(x)$ is irreducible.