Showing a Q-variety is irreducible.

Question

Let $f(x,y)=x^2-2y^2\in{\mathbb{Q[x,y]}}$ then I am not sure how we can show that $V(\{f\})$ as a $\mathbb{Q}$-variety is irreducible.

Answer 1

We have that $V$ is irreducible iff. $I(V)$ is prime in $\mathbb{Q}[x,y]$. (**)

Now $I(V)$ is just the ideal generated by $f$ in this case, and since $\sqrt{2}\not\in\mathbb{Q}$, you can conclude that $f$ is irreducible. You can reference (**) in Knapp's Introduction to Algebraic Geometry.

Admittedly, I am just starting algebraic geometry, and clearly $V_f$ is only an algebraic set when we are working in $\bar{\mathbb{Q}}$, but the one thing I have picked up so far is to be very careful about which field we are in.