Irreducibility of polynomials in two variables

Question

Given $X^2Y-Y^2X+X+Y-1\in \mathbb Q[X,Y]$, how can I check whether this is irreducible or not ?

I know that $(1,0)$ is a root. So, for degree 3 Polynomials over one variable it's enough to find a root, tho..i Guess it's not working here.

Answer 1

Note that $ \mathbf Q[X, Y] = \mathbf Q[Y][X] $. We have a polynomial with coefficients in $ \mathbf Q[Y] $ that is quadratic in $ X $, and over fields of characteristic $ \neq 2 $, we can check the reducibility of such polynomials by simply looking at the discriminant and seeing if it is a square or not. The discriminant is

$$ \Delta = (1 - Y^2)^2 - 4Y(Y-1) $$

This is not a perfect square in $ \mathbf Q[Y] $, because evaluating at $ Y = 3 $ gives $ 64 - 24 = 40 $, which is not a perfect square in $ \mathbf Q $. Therefore, the original polynomial is irreducible.

Answer 2

It's irreducible by Eisenstein's criterion (use the prime $Y-1$)