# Irreducible components of an algebraic set

I am totally stack in how to do this exercise : How can I find the irreducible components of $V(X^2 - XY - X^2 Y + X^3)$ in $A^2$(R)?

Given an algebraic set, what is the consideration I have to make to find them? I know the definitions of components and irreducible components but these don't help in to solve this kind of exercise in the end.

Can anyone suggest a good reference book also?

Thanks

$V(f)$, the zero set of a polynomial $f \in R[x,y]$, is given as set of points such that $f(x,y) = 0$. If this polynomial is reducible, i.e. $f=gh$, the points $(x_0, y_0)$ in $V(f)$ are such that $g(x_0, y_0) = 0$ or $h(x_0, y_0) = 0$. This means that $V(f)$ is a union of $V(g)$ and $V(h)$.
In your case, $f(x,y)=x^2-xy-x^2y+x^3=x(x+1)(x-y)$. Zeros of this polynomial are all the points for which any factor of $f$ is zero. Thus we see that $V(f) = V(x) \cup V(x+1) \cup V(x-y)$. Geometrically, we can imagine this in $\mathbb{R}^2$ as a union of three lines, the y-axis, diagonal line through the origin passing through the first quadrant and parallel to the y axis passing through the point $(-1,0)$.
• Would this change if we consider $A^2(C)$? I don't think we have used anywhere the fact that we are in the field of real numbers