In the very interesting work The real numbers are not interpretable in the complex field about the impossibility of interpreting the real numbers in the complex field by using only the field structure of $\mathbb{C}$ by the great mathematician Joel David Hamkins, he states that the property
(P) Any k-tuples $x \in \mathbb{C}^k$ and $y \in \mathbb{C}^k$ that exhibit the same algebraic equations over $\mathbb{Q}(p_1,\dots,p_n)$ will be automorphic by an automorphism fixing $(p_1,\dots,p_n)$
implies that
(C) there are only countably many isomorphism orbits of $\mathbb{C}^k$.
I have only an elementary knowledge of abstract algebra, and I cannot see why it is so. All that I know is that, being $\mathbb{Q}(p_1,\dots,p_n)$ countable, also $\mathbb{Q}(p_1,\dots,p_n)[T_1,\dots,T_k]$ is countable. Could someone illustrate why the implication asserted by Hamkins holds, please?
Thank you very much in advance for your help.
NOTE. Property (P) was also unknown to me. It has been discussed in my previous post Automorphisms of the Complex Field and Model Theory, where a proof of it has been given by Matthias Klupsch.
For notational convenience, let $K = \mathbb{Q}(p_1,\dots,p_n)$. Now for any tuple $x\in \mathbb{C}^k$, we can look at $I_x = \{f\in K[T_1,\dots,T_k]\mid f(x) = 0\}$. This is an ideal in the polynomial ring over $K$, and to say that $x$ and $y$ exhibit the same algebraic equations over $K$ is to say that $I_x = I_y$.
Now by Hilbert's basis theorem, $K[T_1,\dots,T_k]$ is a Noetherian ring, meaning that every ideal is finitely generated. So we can write $I_x = (f_1,\dots,f_m)$ for some $f_1,\dots,f_m\in K[T_1,\dots,T_k]$. As you noted in your question, $K[T_1,\dots,T_k]$ is countable, so there are only countably many finite tuples $f_1,\dots,f_m$ from $K[T_1,\dots,T_k]$, and hence there are only countably many ideals in $K[T_1,\dots,T_k]$.
Now we can see the implication from (P) to (C): (P) implies that two tuples $x$ and $y$ from $\mathbb{C}^k$ are in the same orbit under automorphisms fixing $K$ if and only if $I_x = I_y$. There are only countably many ideals, so there are only countably many orbits.