Reducible Quadratic form

Question

What is a Reducible/Irreducible quadratic form?. I read about quadratic forms on wikipedia.

I am reading this paper (A.O.L. Atkin, D.J. Bernstein, Prime sieves using binary quadratic forms, Math. Comp. 73 (2004), 1023-1030)

Answer 1

The paper in question concerns binary quadratic forms $Q(x,y)=Ax^2+Bxy+Cy^2$ with coefficients in the rational numbers. Such a form is reducible if and only if there are linear forms $L_1(x,y)=Ux+Vy$ and $L_2(x,y)=U'x+V'y$ with rational coefficients such that $Q=L_1L_2$. Irreducibility (over the rational numbers) is equivalent to the assumption that the discriminant $\Delta=B^2-4AC$ of the form $Q(x,y)$ is not a square in the rational numbers.