A friend of mine has recently drawn my attention to a seemingly shocking result by the mathematician Joel David Hamkins The real numbers are not interpretable in the complex field about the possibility of interpret the real numbers in the complex field by using only the field structure of $\mathbb{C}$.
After a little thought the result seems not so strange to me anymore, but the "elementary proof" given by Hamkins in the quoted page contains two statements which I do not know.
(I) For any $z \in \mathbb{C}$, any two complex numbers transcendental over $Q(z)$ are automorphic in $\mathbb{C}$ by an automorphism fixing $z$.
(II) 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 $(p_1,\dots,p_n)$.
My knowledge of abstract algebra is quite elementary, let us say at the level of Michael Artin's Algebra, and I cannot understand exactly the meaning of these two statements.
Could someone give me some reference where I could find them and their proof?
Thank you very much in advance for your great help.
As was already commented, the first statement is a special case of the second one.
So what is meant $x, y \in \mathbb{C}^k$ satisfying the same algebraic equation over $\mathbb{Q}(p_1,\dots,p_n)$?
It means that for any polynomial $P \in \mathbb{Q}(p_1,\dots,p_n)[T_1,\dots,T_k]$ in $k$ variables we have $P(x) = 0$ if and only if $P(y) = 0$.
This means that the evaluation maps $$\varepsilon_x : \mathbb{Q}(p_1,\dots,p_n)[T_1,\dots,T_k] \to \mathbb{C}, P \mapsto P(x)$$ and $$\varepsilon_y : \mathbb{Q}(p_1,\dots,p_n)[T_1,\dots,T_k] \to \mathbb{C}, P \mapsto P(y)$$ have the same kernel and thus $$\text{im}(\varepsilon_x) \cong \mathbb{Q}(p_1,\dots,p_n)[T_1,\dots,T_k]/\ker(\varepsilon_x) = \mathbb{Q}(p_1,\dots,p_n)[T_1,\dots,T_k]/\ker(\varepsilon_y) \cong \text{im}(\varepsilon_y) $$ via an isomorphism which fixes $\mathbb{Q}(p_1,\dots,p_n)$ and maps $x$ to $y$.
It is a well-known result from Galois theory that a morphism into an algebraic closure extends to that algebraic closure (I believe this should be in Lang's "Algebra"). Thus a morphism $\text{im}(\varepsilon_x) \to \text{im}(\varepsilon_y)$ which fixes $\mathbb{Q}(p_1,\dots,p_n)$ and maps $x$ to $y$ extends to an automorphism of $\mathbb{C}$ which does the same.
Edit:
The situation is a little bit trickier than I assumed.
Any automorphism of a subfield of $\mathbb{C}$ be extended to an automorphism of $\mathbb{C}$. This is Theorem 7 in "Automorphisms of the Complex Numbers" by Paul B. Yale and is a Zorn's Lemma argument similar to the one for extending an isomorphism into $\mathbb{C}$ to the algebraic closure.
So we need to lift our isomorphism $\sigma : \text{im}(\varepsilon_x) \to \text{im}(\varepsilon_y)$ to an automorphism. To do so, let $K$ be the smallest field containing $K_x = \text{im}(\varepsilon_x)$ and $K_y = \text{im}(\varepsilon_y)$. The transcendence degrees of $K/K_x$ and $K/K_y$ are identical, so we find purely transcendent extensions $L_x/K_x$ and $L_y/K_y$ such that $K/L_x$ and $K/L_y$ are algebraic and our $\sigma$ can be lifted to an isomorphism $\sigma: L_x \to L_y$.
Now, by Lang's Thm. 2.8 (or Yale's Thm. 6) we can extend $\sigma$ to an isomorphism from the algebraic closure of $L_x$ to the algebraic closure of $L_y$. But these are identical, since they coincide with the algebraic closure of $K$. We thus get an automorphism of some subfield of $\mathbb{C}$ extending $\sigma$ and this can be extended to an automorphism of $\mathbb{C}$.