I was doing Exercise 3.2 I on Ravi Vakil's notes on Algebraic Geometry: consider the map of sets $\phi: \mathbb{C}^2 \rightarrow \mathbb{A}_{\mathbb{Q}}^2$ defined as follows. $(z_1, z_2)$ is sent to the prime ideal of $\mathbb{Q}[x,y]$ consisting of polynomials vanishing at $(z_1, z_2)$.
I am confused about if the map is well defined or not. Suppose I take the point $(1,1) \in \mathbb{C}^2$. I feel like I could map it to $(x-1, y-1)$ or $(x-y)$ as they are both prime ideals in $\mathbb{Q}[x,y]$... I think I am missing something or not interpreting the question right or something. I would appreciate any help with this. Thanks!
@you-sir-33433 has given you a good answer. But, if you haven't already, it might be worth noting that we can also consider this map as the morphism of affine schemes induced by the natural inclusion $\mathbb Q[x,y] \hookrightarrow \mathbb C[x,y]$ (which makes it immediate that this map is well-defined). To see this, observe that the morphism $\operatorname{Spec}\mathbb C[x,y] \rightarrow \operatorname{Spec}\mathbb Q[x,y]$ associated to inclusion is given by $\mathfrak p \mapsto \mathfrak p \cap \mathbb Q[x,y]$. So, the maximal ideal $\mathfrak m$ ideal of a point in $ p \in \mathbb C^2$ is sent to precisely the set of all elements of $\mathbb Q[x,y]$ which vanish at $p$.