A tool to prove Baby Ax-Kochen principle

Question

I have been reading a lecture notes on model theory of valued fields written by Lou van den Dries. This is the question I have:

It is known in this lecture that:

Let $R$ be a local ring, with $t\in R$ and $n\in \mathbb{N}$ such that: $\text{char}~k=0$ ($k$ is the residue field of $R$), $M=tR$ ($M$ is the maximal ideal of $R$), $t^n\ne0$, $t^{n+1}=0$. Then we have a ring isomorphism $R\cong k[T]/(T^{n+1})$.

By the fact above how can I prove the following:

For each sentence $\sigma$ in the language $L_t:=L\cup\{t\}$, where $L$ is the language of ring and $t$ is a new constant symbol, there is a sentence $\sigma_n$ in $L$ such that for any $R,n,t$ as mentioned above, we have $(R,t)\models\sigma \Leftrightarrow k\models \sigma_n $?

The author said that this is an easy consequence of the above fact; however I cannot see it :(

Answer 1

This is an instance of interpretability. See Wikipedia for a general discussion of the concept.

In our case, we have an interpretation of the structure $(R,t,0,1,+,-,\cdot)$ in the structure $(k,0,1,+,-,\cdot)$, as follows:

• The domain: Each element of $R$ can be represented uniquely as a polynomial in $T$ of degree at most $n$, $a_0 + a_1T + \dots + a_nT^n$, and each polynomial gives an element of $R$. Hence we can encode an element of $R$ by an $(n+1)$-tuple of coefficients, $(a_0,\dots,a_n)$. So we take $k^{n+1}$ as the domain of our interpretation.
• The constant symbols: $0$, $1$, and $t$ are interpreted as the polynomials $0$, $1$, and $T$, which are encoded by the tuples $(0,\dots,0)$, $(1,0,\dots,0)$, and $(0,1,0,\dots,0)$ in $k^{n+1}$ (note that all coordinates of these tuples are constant symbols in $k$).
• The operations: Addition and multiplication in $R$ is performed as addition and multiplication of polynomials, mod $T^{n+1}$. On tuples, this corresponds to coordinate-wise addition for $+$, and a rather more complicated (but still easy to write down) formula for $\cdot$.

Now to show that any sentence in $L_t$ can be translated to a sentence in $L$, do induction on the complexity of the sentence. Replace each term with an $(n+1)$-tuple of terms according to the translation above. Replace $t = t'$ by $\bigwedge_{i=0}^n (t_i = t'_i)$, where the $t_i$ are the $n+1$ components of $t$, and similarly for $t'$. Replace a quantifier with a block of $n$ quantifiers, one for each coefficient. Do you see how to fill in the details?