Irreducible polynomials in $\mathbb{C}[X,Y]$

Question

I have the polynomial $X^2+Y^2-1$ in $\mathbb{C}[X,Y]$.

Is this irreducible? If not, how do I factorize it?

Should I handle this the same as if it were $\mathbb{R}[X,Y]$, or should I do it differently?

Answer 1

Let $R$ be a UFD of characteristic $\neq 2$. Then $X^2+Y^2-1 \in R[Y][X]$ has no root in $R[Y]$. Indeed, a root $P \in R[Y]$ would satisfy $P^2=1-Y^2=(1+Y)(1-Y)$, which is impossible since $1+Y$ and $1-Y$ are non-associated prime elements in $R[Y]$. Thus, $X^2+Y^2-1$ is irreducible in $R[Y][X]$, being a polynomial of degree less than $4$.