Let $K$ be a field and let $R = K[X_1, ..., X_n]$.
For $K = \mathbb{R}, n = 2$, and $V = {\{(x_1,x_2) \in K^2 : |x_1| = |x_2|}\}$.
How do I determine if there exists an ideal $I$ of $K[X_1, X_2]$ such that $V(I) = V$, where $V(I)$ denotes the variety of $I$.
I know that $V(I) = {\{ (x_1, x_2) \in K^2 : f(x_1, x_2) = 0 \forall f \in I}\}$
So in the case where K = $\mathbb{R}$, the ideal $I = <X_1^2 - X_2^2>$ would work I believe.
How about for $K = \mathbb{C}$? Does one exist? I'm not sure where t start here!
In $K=\Bbb C$ we have $I=(X_1^2-X_2^2)=(X_1-X_2)(X_1+X_2)=I_1I_2$, so $V(I)=V(I_1I_2)=V(I_1)\cup V(I_2)$