I am very confused about the Lefschetz Principle. I read the Tarski Principle, but I am not acquainted with logic. Is there a statement more close to the language of field theory?
Most of all, I would like to see concretely how a field $k$ of characteristic zero, containing $\mathbb C$, can be embedded into $\mathbb C$. For example:
Let $x_1,\dots,x_n$ be transcendental elements over $\mathbb C$. How to construct an embedding of $k=\mathbb C(x_1,\dots,x_n)$ into $\mathbb C$?
Is it possible to construct a similar embedding starting with a field $k/\mathbb C$ of infinite transcendence degree over $\mathbb C$?
If we start with $k$ algebraically closed of finite transcendence degree over $\mathbb C$, should we get an isomorphism $k\cong \mathbb C$ after the embedding constructed in 1?
In addition, I would like to know how one can use this result in Algebraic Geometry: if we work with algebraic varieties over $k$, they are all determined by finitely many coefficients in $k$. What kind of statements can we prove just "as if" they were defined over $\mathbb C$? What happens with a scheme over $k$ which cannot be covered by finitely many affine schemes? In fact, as in 2, I'm wondering if some finiteness condition is essential for performing "reduction steps" via Lefschetz Principle.