This is Fulton, Algebraic Curves exercise 1.31:
We have to find irreducible components of $V(f) =V(y^2-xy-x^2y+x^3)$ in $\mathbb{A^2({\mathbb{R}})}$ and $\mathbb{A^2({\mathbb{C}})}$.
My attempt: Is to write this polynomial as = $(y-x^2)(y-x)$ And so we have $V(f)= V(y-x^2) \cup V(y-x)$ My guess is that it would be the same set of points in both $\mathbb{A^2({\mathbb{R}})}$ and $\mathbb{A^2({\mathbb{C}})}$.
Is my guess correct? Any alternate approach or output on my idea will be appreciated, thanks.
$V(f)$ denotes the set of all points at which $f$ is zero in $\mathbb{A^2({\mathbb{k}})}$, $k$ a field.