How complicated is the theory of $\mathbb Q_p$?

Question

Let $p=7$, say. How complex is the theory of the field $(\mathbb Q_p,0,1,+,\cdot)$ ? One extreme would be that the theory is like $\mathbb R$: decidable

... admits quantifier elimination, is o-minimal. Do I need to include $\;<\;$ for $\mathbb R$ for these? Or is it enough that $<$ is definable in $(\mathbb R, 0,1,+,\cdot)$ ?

The other extreme would be that $\mathbb Z$ is definable in $\mathbb Q_p$, so the theory is as complex as the theory of $\mathbb Z$.

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I thought of this when reading a recent question, asking whether $\mathbb C$ and $\mathbb C_p$ admit a Borel-measurable isomorphism.

Answer 1

There is a strong analogy between the model theory of $\mathbb{R}$ and the model theory of $\mathbb{Q}_p$. Here's a brief history of some of the main results (I've necessarily left a lot out, so I encourage others to edit and add to the list):

• In the '60s, the theory of $\mathbb{Q}_p$ was shown to be decidable, independently by Ershov (On the elementary theory of maximal normed fields, 1965) and Ax and Kochen (Diophantine problems over local fields. III. Decidable fields, 1966).
• In the '70s, Macintyre proved that the structure $(\mathbb{Q}_p;0,1,+,-,\times,^{-1},V,(P_n)_{n\in \mathbb{N}})$ has quantifier elimination, where $V$ is a unary predicate naming the valuation ring, and each $P_n$ is a unary predicate naming the set of $n^{\text{th}}$ powers. Like $\mathbb{R}$ (and like every non-algebraically-closed field, by another result of Macintyre's - see below for more on this), $\mathbb{Q}_p$ does not have quantifier elimination in the language of rings. (On definable sets of $p$-adic fields, 1976).
• In the '80s, Denef proved a cell decomposition theorem for $\mathbb{Q}_p$, analogous to Tarski's cell decomposition for real closed fields. (See The rationality of the Poincaré series associated to the p-adic points on a variety, 1984, and p-adic semi-algebraic sets and cell decomposition, 1986).
• In the '90s, Haskell and Macpherson defined a $p$-adic analogue of o-minimality, called "P-minimality" (A version of o-minimality for the p-adics, 1997). They did not obtain a cell decomposition theorem for general P-minimal structures, and since then there have been a number of papers about cell decomposition in the P-minimal setting, usually under additional hypothesis, such as definable Skolem functions.
• It follows easily from cell decomposition that the theory of $\mathbb{Q}_p$ is NIP, and this was probably folklore for some time. The dp-rank is a measure of complexity for NIP theories. In the 21st century, Dolich, Goodrick, and Lippel showed that the theory of $\mathbb{Q}_p$ is dp-minimal, meaning that it has dp-rank 1 (dp-minimality: basic facts and examples, 2011). The theory of real closed fields is also dp-minimal, as is any o-minimal theory.

Regarding your side question: most model-theoretic notions are impervious to the exact choice of language, as long as the languages considered are interdefinable. This includes notions like decidability, dp-minimality, and NIP. Characterizations of definable sets, like o-minimality, are mostly language-independent, except that e.g. the definition of o-minimality requires the symbol $<$ to be in the language. On the other hand, quantifier elimination is highly sensitive to the choice of language.

So for $\mathbb{R}$, decidability (and dp-minimality, and NIP) holds in the language of rings, quantifier elimination does not hold in the language of rings, and it doesn't even make sense to ask about o-minimality in the language of rings.

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P.S. Above I alluded to a theorem of Macintyre on fields admitting quantifier elimination. The original reference is On $\omega_1$-categorical theories of fields, 1971, where Macintyre proves that the following are equivalent for infinite fields (in the language of rings):

1. $K$ is algebraically closed
2. $\text{Th}(K)$ is $\aleph_1$-categorical
3. $K$ is totally transcendental ($\omega$-stable)
4. $\text{Th}(K)$ has quantifier elimination

A simpler argument for the equivalence of (1) and (4) was given by Macintyre, McKenna, and van den Dries in Elimination of quantifiers in algebraic structures, 1983. The main results are the following:

• If a field admits quantifier elimination in the language of rings, it is algebraically closed.
• If an ordered field admits quantifier elimination in the language of ordered rings, it is real closed.
• If a non-trivially valued field admits quantifier elimination in the language of of rings together with a binary relation symbol for $v(x)\leq v(y)$, it is an algebraically closed valued field.
• If a $p$-field (see the paper for the definition) admits quantifier elimination in the Macintyre language described above (with $V$ and $(P_n)_{n\in \mathbb{N}}$), it is $p$-adically closed (elementarily equivalent to $\mathbb{Q}_p$).

Answer 2

This is not by any means a complete answer, but I'd like to record a few facts that hopefully address at least some of your questions. Fix a prime $p$ and let $T$ denote the complete theory ​of $\mathbb{Q}_p$ in the language of rings; for convenience, I'll assume that $p$ is odd. (The case when $p=2$ is similar, but some details are a bit different.) Define a formula $\varphi(v)\equiv\exists w[w^2=pv^2+1]$. Rather remarkably, this gives a definition of $\mathbb{Z}_p$ in $\mathbb{Q}_p$; this is a good exercise using Hensel's lemma. So this is immediately enough to tell us that $T$ does not have elimination of quantifiers; indeed, the quantifier-free formulas in $T$ are just Boolean combinations of polynomial equations, so every quantifier-free definable subset of $\mathbb{Q}_p$ is either finite or cofinite. Since $\mathbb{Z}_p$ is neither, we cannot have QE in the pure language of rings.

There are however several natural ways of enriching the language to obtain QE. For example, the following is due to Angus Macintyre. Let $L_{\text{mac}}$ be the union of the language of rings with a predicate symbol $\mid$ and predicate symbols $P_n$ for each $n\in\omega$. Let $v$ be the valuation on $\mathbb{Q}_p$, and realize $\mid$ in $\mathbb{Q}_p$ by taking $a\mid b$ to hold if and only if $v(a)\leqslant v(b)$; further realize each $P_n$ in $\mathbb{Q}_p$ by taking $P_n(a)$ to hold if and only if $a$ is an $n$-th power. Then, by a theorem of Macintyre's, $\mathbb{Q}_p$ admits elimination of quantifiers in the language $L_{\text{mac}}$. With some work, one can use this QE result to show that $T$ has something called the "no independence property", aka NIP. If you are unfamiliar with this notion, it is basically a kind of "tameness" condition, prohibiting the existence of certain "bad" combinatorial configurations within models of $T$. In particular, no NIP theory can interpret an IP theory. The theory of the ring $\mathbb{Z}$ has IP, so this fact is enough to show that $\mathbb{Z}$ is not only not definable in $\mathbb{Q}_p$, but also not even interpretable in $\mathbb{Q}_p$.

Okay, if I'm not misunderstanding, you then ask whether $\mathbb{Q}_p$ might be o-minimal. This is a bit of a "type error", as o-minimality is something that applies to totally ordered structures. I don't know of any "natural" total ordering on $\mathbb{Q}_p$, and, in contrast with $\mathbb{R}$, there certainly does not exist an ordering on $\mathbb{Q}_p$ compatible with the field structure; for example, $1-p$ admits a square root in $\mathbb{Q}_p$. There is however a generalization of o-minimality called dp-minimality, and it apparently is the case that $T$ is dp-minimal; I am not familiar with dp-minimality, so perhaps someone on the site who has more experience with that notion could provide further insight.

Finally, let me just give a few references for further reading. Disclaimer: the model theory of valued fields is a very deep field, and I only know a small amount of it. Here and here are some notes of David Marker and Zoe Chatzidakis, and Alex Kruckman's undergraduate thesis gives a very nice exposition of the Ax-Kochen theorem. Finally, if you are interested in learning about things like NIP theories in the context of valued fields, you may want to check out Appendix A.2 of Pierre Simon's book on NIP theories, which gives an overview of many of the things I discuss above. Enjoy!