Let $k$ and $l$ be algebraically closed fields. Let $K/k $ and $L/l$ be finitely generated extensions. Suppose $K$ and $L$ are elementarily equivalent. Then
- $k$ and $l$ have the same prime field $F$;
- $\text{td}(K/k) = \text{td}(L/l)$
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Let $(t_1, \ldots, t_d)$ be a separable transcendence basis of $K/k$.
Notice: Unlike in the arithmetic case, we cannot in general write $K = F(t_1, \ldots, t_d)[x]$. (Why?)
Suppose $K = k(t_1, \ldots, t_d)[x]$ and $P(\underline{T}) \in k[\underline{T}]$ is an irreducible polynomial such that $P(t_1, \ldots, t_d, x) = 0.$ (In particular, $K$ is the function field of the irreducible variety $\text{Spec} k[\underline{T}]/(P)$.)
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Cutting out $x$ by a polynomial with coefficient in $F$.
As $P$ does not have coefficients in $F$, we cannot use $P$ to cut out $x$. Our idea is to try to replace $P$ with a polynomial $Q$ with coefficients in $F$.
Let $\alpha_i$'s be the coefficients of $P$, then $P$ is defined over $F[\underline{\alpha}] \subset k$, i.e. we can view it as an element of $F[\underline{\alpha}][\underline{T}].$ Now $F[\underline{\alpha}= F[\underline{Z}]/(\underline{f})$ for some finite system $\underline{f}$ of relations $f_j$ of the variables $Z_i$.
So we can view $P$ as the equivalence class of an irreducible (why?) polynomial $Q \in F[\underline{Z}, \underline{T}].$
So we can view $P$ as the equivalence class of an irreducible (why?) polynomial $Q \in F[\underline{Z}, \underline{T}].$
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Cutting out $\underline{\alpha}, \underline{t}, x$ by a formula.
Let $\Psi(\underline{\zeta},\underline{\xi}, z)$ be the conjunction of
- $\Xi (\underline{\xi})$, i.e. $\underline{\xi}$ is a transcendental basis;
- $\underline{f}(\underline{\zeta})$;
- $P(\underline{\zeta}, z).$
Define $\iota: K \to L$.
As $\Psi$ has roots in $K$ and $K \equiv L$, $\Psi$ must have roots $(\underline{\beta}, \underline{u},y)$ in $L$. Thus we define $\iota: F[\underline{\alpha}, \underline{t}, x] \to L$ by
$\alpha_i \mapsto \beta_i$, $t_i \mapsto u_i$ and $x \mapsto y$.
Then as $\underline{u}$ is a root of $\Xi$, it must be a separable transcendence basis of $L/l$.
What is the problem?
We want to extend $\iota$ to a function field embedding $K/k \to L/l$ where $K = k[t_0, \ldots, t_d][x]$. To this end, it suffices to extend
$\iota': R=F[\underline{\alpha}] \to F[\underline{\beta}]$ to a field extension $k \to l$.
The problem is:
- Image of $\iota'$ might not lie in $l$
- $\iota'$ might not be injective
Even when these two problems are resolved, we could only guarantee an embedding of $k_0 = \text{Frac} R$ into $l$. This would extend to a function field embedding $K_0/k_0 \to L/l$ where $K_0 = k_0(\underline{t},x).$
We do not know if (2) can be resolved, but (1) can be resolved by replacing $\Psi$ with a more complicated formula. This is because the constant are definable. In other words, there exists a formula $\Theta(a)$ such that $\Theta(a)$ holds in $K$ iff $a \in k$.
Take $\Psi'$ to be $\Psi(\underline{\zeta},\underline{\xi}, x) \vee \Theta(\zeta_i).$
Take $\Psi'$ to be $\Psi(\underline{\zeta},\underline{\xi}, x) \vee \Theta(\zeta_i).$
Then the roots $\beta_i$ have to all satisfy $\Theta(\beta_i)$ in $L$ and thus must lie in $l$.
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