Ax-Kochen theorem

Question

I'm following the proof of Ax-Kochen theorem by this paper: http://arxiv.org/abs/1308.3897 I have two question in the proof of Ax-Kochen Principle:

• In the page 49, I didn't understend the argumentation to conclude that residue class field and value group of $F$ are $\aleph_1$-satured in their respectives languages. I can't proof that the translation of the type to valued field $F$ is satisfatible.

• In the page 52, there is the following implication: $\upsilon_F(a-b) < \upsilon_F(x-a) \rightarrow f_1(\upsilon_F(a-b)) < f_1(\upsilon_F(x-a))$. Why we can do that? Its know that $f_1$ is a partial elementary bijection of groups (not linearly ordered groups) between $\upsilon_F(F_1^*)$ e $\upsilon_G(G_1^*)$. In the next page, the same implication is used to calculate the valoration of a polynomial.

Answer 1

Glad to see someone's reading my undergrad thesis!

First question: You say your trouble is in showing that a type in the language of the residue field or value group translates to a consistent type in the language of the valued field. The following rewording of compactness may help you:

A set of formulas $\Phi(\overline{x})$ in the free variables $x_1,\dots,x_n$ with parameters from some model $M$ is consistent with $T = \text{Th}(M)$ if and only if every for every finite set $\phi_1,\dots,\phi_m\in \Phi$, the formula $\bigwedge_{i = 1}^m\phi_i(\overline{x})$ has a realization in $M$. Why? By compactness, $\Phi$ is consistent iff it every finite subset has a realization in some model of $T$. But since $T$ is complete, $\bigwedge_{i = 1}^m\phi_i(\overline{x})$ has a realization in some model of $T$ iff $\exists \overline{x} \bigwedge_{i = 1}^m\phi_i(\overline{x})\in T$ iff $\bigwedge_{i = 1}^m\phi_i(\overline{x})$ has a realization in $M$.

Now in our particular case, let $K$ be the valued field and $k$ the residue field. A type $p$ in the language of the residue field translates to a set of formulas $\Phi$ in the language of the valued field (replacing each parameter with a representative for its residue class). Every finite subset of $p$ has a realization in $k$, which (taking a representative) gives a realization of the corresponding finite subset of $\Phi$ in $K$. Hence $\Phi$ is consistent. By saturation, it has a realization in $K$, the residue of which is a realization of $p$ in $k$. The same argument works for the value group.

The fact that we can talk about the residue field (or the value group) in the theory of valued fields is an example of the notion of interpretation in model theory. In general, if a theory $T$ in a language $L$ interprets a theory $T'$ in a language $L'$ and $M\models T$, the interpretation gives us a model $M'\models T$. Not only do formulas of $L'$ correspond to formulas of $L$, types over $M'$ correspond to (partial) types over $M$, and if $M$ is ($\kappa$-)saturated, so is $M'$.

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Second question: $f_1$ preserves the order on the value group by definition:

We will write $f_1:F_1\leftrightarrow G_1$ if and only if $f_1$ is an isomorphism between $F_1$ and $G_1$, and $f_1$ restricted to the value group of $F_1$ is a partial elementary bijection between $\mathfrak{v}_F(F_1^*)$ and $\mathfrak{v}_G(G_1^*)$ as subsets of $\mathfrak{v}_F(F^*)$ and $\mathfrak{v}_G(G^*)$.

The definition of "partial elementary bijection" is unfortunately hidden in Theorem 4.3.6, but in particular a partial elementary bijection preserves the symbols in the language, and the ordering on the value group is included in the language of valued fields.