Show $xy-2$ is irreducible in $\mathbb{R}[x,y]$

Question

I figured using the Eisenstein criterium but my book says nothing about multiple variables. It is probably very trivial but just a lack of knowledge on how this stuff works with multiple variables.

Answer 1

Since your polynomial has degree $1$ in $x$, in any factorization as $p(x,y) q(x,y)$ one of the terms will have degree $1$ in $x$ and the other degree $0$ in $x$. Similarly for $y$. Since we don't allow one of the terms to be degree $0$ in both $x$ and $y$, the only possibility would be $(a x + b)(c y + d)$.

But this is easy to rule out: there is no constant value of $x$ or $y$ that makes the polynomial $0$.