These are problems from Miles Reid's book "Undergraduate algebraic geometry"
Let $k$ be an algebraically closed field.
Question 1.
Let $I= (xy,xz,yz) \subset k[x,y,z]$. I want to find $Z(I)$ ok it is clear that $I$ is the union of the three coordinate axes, now I want to prove rigorously each coordinate axis is closed and irreducible. Can we simply say:
For example the z-axis is homeomorphic to $\mathbb{A}^{1}$. But any homeomorphism preserves closed sets and irreducible sets so since $\mathbb{A}^{1}$ is algebraic (i.e closed) and irreducible we are done.
Question 2:
Let $I=(x^{2}+y^{2}+z^{2},xy+xz+yz)$. Identify $Z(I)$ and $I(Z(I))$.
I'm confused with this one because here in case the field is $\mathbb{R}$ then we get $x=y=z$ however $\mathbb{R}$ is not algebraically closed so how do we proceed?
Now I was thinking in using the identity $(x+y+z)^{2}=x^{2}+y^{2}+z^{2}+2(xy+xz+yz)$ from here we get that $x+y+z=0$. So it seems $Z(I)$ is the set of all points in the plane $x+y+z=0$. Is this wrong? But what if the underlying field has characteristic equal $2$?
Finally how do we find $I(Z(I))$?