What is the ideal of a point in algebraic geometry?

Question

I found a problem as follows: Find the ideal of a point $z$, denoted by $\mathfrak j_z\subset\mathbb Q[X,Y]$, and its conjugates in $\mathbb C^2$ as $z=(\sqrt{2},\sqrt{3})$. I tried to Google but found no explanation what the question asks.

Answer 1

For any field $k$, and any subset $S\subseteq k^n$, the ideal $I(S)\subseteq k[x_1,\ldots,x_n]$ is defined to be $$I(S)=\{f\in k[x_1,\ldots,x_n]:f(z)=0\text{ for all }z\in S\}$$ See this discussion on Wikipedia for example.

The point $z=(\sqrt{2},\sqrt{3})\in\mathbb{C}^2$ has its ideal $$I(\{z\})=\{f\in\mathbb{C}[x,y]:f(\sqrt{2},\sqrt{3})=0\}$$ and the symbol $\mathfrak{j}_z$ seems to be this author's notation for the ideal $$\mathfrak{j}_z=I(\{z\})\cap\mathbb{Q}[x,y]=\{f\in\mathbb{Q}[x,y]:f(\sqrt{2},\sqrt{3})=0\}$$