p-adic fields for beginners

Nothing in this post is particularly new, but I felt that it should be written in HTML somewhere. It’s just a collection of standard facts about p-adic fields and counterexamples. I’m new to this blog-posting thing, so let me know if I’ve committed a Web 2.0 faux pas.

It’s well-known that the ring of integers admits infinitely many structures of a metrized topological ring. The standard archimedean metric \Vert x - y \Vert_\infty = | x - y | is the one we learn in grade school, and for each prime p, there is also a p-adic metric \Vert x - y \Vert_p = p^{-v_p(x-y)}, where v_p(x-y) is the largest n such that p^n | (x-y) if it exists, and +\infty if x = y. The p-adic metrics (unique up to normalization) are called ultrametrics, because they satisfy the following strengthening of the triangle inequality: \Vert x - z \Vert \leq max( \Vert x - y \Vert, \Vert y - z \Vert). It implies that all triangles are isosceles, and for any two balls with nonempty intersection, one contains the other. Each of the above metrics can be uniquely extended to \mathbb{Q}, and completing yields the field of real numbers in the archimedean case, and the p-adic fields \mathbb{Q}_p for each prime p. These fields are useful in number theory for a variety of reasons, e.g., they have much simpler arithmetic structure (Diophantine equations, and more generally, first order sentences are decidable, and the Galois groups of these fields are pro-solvable), and the complete topology allows questions to be answered using analytic techniques, such as Newton’s method (aka Hensel’s lemma).

\mathbb{Q}_p has a unique maximal compact subring, called \mathbb{Z}_p, the ring of p-adic integers, which is both the completion of \mathbb{Z} under the p-adic metric and the subset of elements of \mathbb{Q}_p with non-negative valuation. \mathbb{Z}_p in turn has a unique maximal ideal, p\mathbb{Z}_p, which generates the topology, and the quotient, called the residue field, is the field with p elements. The topology on \mathbb{Z}_p is fractal in nature, and in fact, \mathbb{Z}_2 is homeomorphic to the Cantor set. Elements of \mathbb{Q}_p can be conveniently written as Laurent series in p with coefficients in the residue field, but addition and multiplication are more complicated than in the Laurent series field \mathbb{F}_p((t)) = \{ \sum_{n \gg -\infty} a_n t^n | a_n \in \mathbb{F}_p \} because of nontrivial carries. You will typically see this presentation explained as writing rationals in base p, but letting the expansion extend infinitely far to the left. Standard examples include -1 = \ldots 1111 = \sum_{n \geq 0} 2^n \in \mathbb{Z}_2 and 2/15 = \ldots 313131.4 \in \mathbb{Q}_5. We cannot let the expansion extend infinitely far in both directions, because multiplication becomes hard to define.

Now that we understand \mathbb{Q}_p, I will spend the rest of the post discussing some distinguished extensions. We have the following strict inclusions:

\mathbb{Q}_p \subset \mathbb{Q}_p^{nr} \subset \overline{\mathbb{Q}}_p \subset \mathbb{C}_p \subset \Omega_p

The last three fields are ring-theoretically isomorphic to \mathbb{C}, assuming the axiom of choice, and the existence of an embedding is employed in Deligne’s theory of weights. There are additional extensions of interest, such as \mathbb{Q}_p^{tame}, a maximal tamely ramified extension, and \mathbb{Q}_p^{ab}, a maximal abelian extension, but they are beyond the scope of this post. I’ll add some references at the end.

\mathbb{Q}_p^{nr} is known as a maximal unramified extension of \mathbb{Q}_p. It is given by adjoining all nth roots of unity, for (n,p)=1. The elements of this field can be represented by Laurent series in p, with coefficients in \overline{\mathbb{F}_p}, although this is an imprecise way to describe it. There are two standard ways to remedy this: First, one can say that the coefficients are Teichmüller lifts of elements in \overline{\mathbb{F}_p}, in other words, prime-to-p roots of unity or zero. Alternatively, one can work with Witt vectors, representing the coefficients as (x,0,0,\ldots), x \in \overline{\mathbb{F}}_p. Witt vectors become very messy if you try to do anything explicit (Lenstra once remarked, “the formulas do not fit in the head of a civilized mathematician of the twenty-first century.”).

\overline{\mathbb{Q}}_p is an algebraic closure of \mathbb{Q}_p. Typical elements not in \mathbb{Q}_p^{nr} are roots of p, p-power roots of unity, and more exotic examples like roots of x^p - x - 1/p (discovered by Chevalley). Their expansions in powers of p do not necessarily have exponents that form a discrete subset of the rationals. Bjorn Poonen’s undergraduate thesis has \sqrt{-1} \in \overline{\mathbb{Q}_2} partially expanded in powers of two, and the set of exponents with nonzero coefficients seems to have order type \omega^\omega, although I don’t know if anyone has bothered to prove it. Kedlaya’s third paper on the arXiv describes which series lie in this field, using twist-recurrence relations, and shows that the aforementioned order type is the largest possible for algebraic elements.

\mathbb{C}_p is the p-adic completion of \overline{\mathbb{Q}}_p. One can generate transcendental elements in \mathbb{C}_p by summing a rapidly decaying sequence of elements of increasing degree such as p^{\frac{n!+1}{n}}, and the proof that they are transcendental is based on the fact that the action of the Galois group preserves distances. Gouvea’s undergraduate textbook has a detailed construction of a Cauchy sequence in \mathbb{Q}_p^{nr} that doesn’t converge in \overline{\mathbb{Q}}_p. \mathbb{C}_p is one of the most commonly used base fields for p-adic analysis and geometry, since it is complete and algebraically closed while still having a countable dense subset. A lot of classical Banach space theory works well here, e.g., open mapping theorem, closed graph theorem.

\Omega_p is the spherical completion of \mathbb{C}_p, although I don’t think the notation is completely standardized. A spherically complete metric space is one for which every sequence of nested balls of finite radius has nonempty intersection. This implies completeness, since that is the special case in which the radii of the balls approach zero, and it is necessary for the Hahn-Banach theorem to hold. Rather surprisingly, \mathbb{C}_p is not spherically complete, and in fact, one can reasonably say that most nested sequences of balls in \mathbb{C}_p have empty intersection. One example (pointed out to me by Bjorn) is given by taking the balls of the form O(p^{-1/2}), p^{-1/2} + O(p^{-1/3}), p^{-1/2} + p^{-1/3} + O(p^{-1/5}), p^{-1/2} + p^{-1/3} + p^{-1/5} + O(p^{-1/7}), \ldots, where the exponents are reciprocals of primes. The radii decrease to 1, but there is no Cauchy sequence of algebraic elements that begin with this sort of expansion.

How do we write elements of \Omega_p? The answer quite simple, and is found in Poonen’s undergraduate thesis. The elements are exactly those power series \sum_{r \in \mathbb{Q}} a_r p^r with coefficients given by Teichmüller representatives of \overline{\mathbb{F}}_p, such that the set of exponents with nonzero coefficients forms a well-ordered subset of the rationals. Apparently, the hardest part of the proof was showing that this set was closed under addition and multiplication.

One might ask if there are natural extensions of \Omega_p that don’t add geometry (e.g., taking rational functions shouldn’t count). The answer is no. There is a notion of maximally complete nonarchimedean field, because if we fix the data of the characteristic, the residue field, and the value group, then there is a maximal field with respect to those properties (assuming the data can actually come from a field), and it is unique up to nonunique isomorphism [***Update Aug. 2015: Kevin Buzzard emailed me to point out that this is not true in the generality stated here.  Kaplansky showed in 1942 that we have uniqueness when the residue characteristic is zero, and in many cases of characteristic p (e.g., the value group is p-divisible and the residue field contains roots for a certain class of polynomials).  However, in 1945, Kaplansky produced examples of non-uniqueness in positive residue characteristic.] However, there is a theorem asserting that maximally complete fields and spherically complete fields are the same thing. In particular, given any proper containment of fields K \subset L such that they both yield the same data, the smaller field cannot be spherically complete.

The whole framework above more or less extends to the fraction field of any complete discrete valuation ring, such as \mathbb{C}((t)). Since the field of complex numbers is algebraically closed, the Laurent series field has no unramified extensions. The algebraic closure is the field of Newton-Puiseaux series given by \bigcup_{n > 0} \mathbb{C}((t^{1/n})), which is not t-adically complete, and its completion is not spherically complete, closely mirroring the local field case. The absolute Galois group of \mathbb{C}((t)) is \widehat{\mathbb{Z}} = \varprojlim \mathbb{Z}/n\mathbb{Z}, isomorphic to that of \mathbb{F}_p, but instead of a Frobenius, there is the monodromy operator \sum a_r t^r \mapsto \sum e^{2 \pi i r} a_r t^r. You can think of the spectra of both fields as really small circles, so the Zariski topology only sees a point, while the étale topology is a strong enough magnifying glass to see the finer structure. This picture is loosely connected to the arithmetic topology view of Spec \mathbb{Z} as a three-ball with the primes forming embedded knots.

Further reading:

Schneider has a relatively new book on nonarchimedean functional analysis. I had expected it to be light reading, but I was wrong.

Bosch, Güntzer, and Remmert have a book that introduces nonarchimedean analytic geometry. You should skip the first six chapters, unless you really like Japanese rings.

The Arizona Winter School held a p-adic geometry workshop this March, and Conrad has excellent notes on the foundations of rigid analytic spaces. Most of the speakers have excellent notes.

I don’t recommend Tate’s Inventiones article on rigid analytic spaces, although it has a cute joke about following Grothendieck fully and faithfully.

For information on other interesting algebraic extensions of \mathbb{Q}_p, look in Serre’s Local Fields, and Serre’s chapter VI in Cassels and Frohlich. In fact, you might as well get everything you can find by Serre, because he writes beautifully.

46 thoughts on “p-adic fields for beginners

  1. Scott, it’s definitely not a faux pas to write about stuff that is “somewhere else” or that “everybody knows” since this is often the hardest stuff to find a good reference for. I recommend you keep up the trend.

  2. One way in which the analogy between the etale topology of F_p and C((t)) breaks down is in the eigenvalues of monodromy. If you have a variety X over F_p, then the Galois action on the l-adic cohomology has interesting eigenvalues. Deligne got a Fields Medal for proving something them. On the other hand, over C((t)), monodromy is quasi-unipotent. So the eigenvalues there are not really interesting.

  3. On the contrary, I think that the fact that the monodromy is quasi-unipotent is a very interesting fact about the eigenvalues!

    Kedlaya has a series of papers explaining and proving the p-adic analogue of the quasiunipotency result. (The key phrase is “Semistable reduction”.) They are well beyond my ability to understand in detail, but one thing that I did follow is that the direct analogue of the monodromy is not the Galois action but rather parallel transport along a connection. (Even at the level of my understanding, I am cheating here, but I hesitate to be more specific about something that I don’t really understand.) However, this parallel transport commutes with the Galois action and can therefore be used to restrict the eigenvalues of Frobenius.

  4. Of course I agree with you, Ben. I’m just being a bit snarky.

    Post on, Scott. My entire weblog consists of stuff that’s around elsewhere, just organized all in one place and explained the way I think of it.

  5. This sort of cuts to the heart of most math weblogs, right? That we (mostly) aren’t saying new things, we are just putting our own spin on things. I for one have been detered when I realize just how much I want to say is right there on the internet. Hell, at this point, 50% of the stuff I would write about is already sitting inside the archive of This Weeks Finds.

    Still, I think its almost always worthwhile to write something again. I know I rarely understand a subject until I learn it for the third or fourth time. Maybe your post reminds someone that they never learned a subject as well as they want and they always meant to go back and sort things out; or maybe they just had a terrible teacher who defined something in the worst imaginable way. Besides, I somewhat shield myself from concerns about the audience’s well-being by reminding myself that I benefit from writing the post as much as anyone benefits from reading it.

    There’s another reason why I am ok with blog posts about existing math, and it has to do with my high-concept vision for the future of math communication. As the arxiv and other online methods for publishing papers grows in ubiquity, the real problem will be (and already is) finding which papers are worth reading. I see a potential solution as a handful of respected blogs in each field, which mention noteworthy papers just put on the arxiv several times a month. The blogs could even be dedicated towards this purpose, an ‘online journal’ of whatever, with a panel of editors running things.

  6. One great thing about reading old mathematics on blogs, is that (like watching people teach) it tells you a lot about the mathematical taste of other mathematicians. I think learning what other mathematicians like and don’t like is very important in terms of guiding ones own taste and figuring out which research programs would be of more general interest.

  7. Joel: I have a ring R of elements with nonnegative valuation and a canonical homomorphism to its residue field F = \overline{\mathbb{F}_p} given by killing the maximal ideal, i.e., positive valuation elements. If R were a power series ring, then it would be an F-algebra, and this map would have a uniquely defined ring-theoretic section. In the case of mixed characteristic here, it is impossible to make a section that is an additive homomorphism, so the Teichmüller lift is the next best thing, which is a morphism of multiplicative monoids, and it is uniquely defined. The units of F form an ind-cyclic torsion group (I think group theorists say “quasicyclic”), so the image is given by prime-to-p roots of unity and zero.

    The lift works whenever you have an algebra over the Witt vectors of a perfect field, by the universal property of strict p-rings. You can find more discussion at the end of chapter 2 of Corps Locaux (grain of salt – quoting from memory). This fails rather miserably when the residue field is not perfect, but I’ve forgotten the precise functor that fails to be representable.

  8. David, I think we agree. I would put it this way: it is a very interesting fact that the eigenvalues of local monodromy are uninteresting.

  9. James and David, you’ve successfully plunged this discussion into waters well over my head. I’ll try to clarify a couple things, though.

    Schmid showed that monodromy is quasi-unipotent for any variation of polarized Hodge structures (not just a family of varieties), so the deRham cohomology has some structure that is missing from the l-adic étale cohomology of an \mathbb{F}_p-variety. Note, however, that Grothendieck proved a quasi-unipotence theorem for l-adic monodromy of varieties over the fraction field of a Henselian local ring with residue field of characteristic p. In the sketchy picture I gave above, I guess the "circles" coming from punctured Henselian traits are somehow bigger than the \mathbb {F}_p circles, but roughly the same size as the rigid annuli with Frobenius structure in the André-Mebkhout-Kedlaya proofs of the Crew-Tsuzuki conjecture.

    I can’t say much about semistable reduction, although there is something deep and beautiful going on there. If a family of complex varieties has semistable reduction at the special fiber, then you get a limit mixed Hodge structure, and a vanishing cycles complex with a nilpotent action of the logarithm N of the monodromy operator from the Gauss-Manin connection on the relative deRham cohomology. I think Steenbrink showed that the weight filtration on the mixed Hodge structure coincides with the filtration coming from N (I think this is the same N that shows up in Weil-Deligne representations). Deligne’s monodromy-weight conjecture is an l-adic analogue of this, where weights here describe the archimedean absolute values of eigenvalues of Frobenius lifts to local fields. There is also a p-adic version conjectured by Mokrane, about Hyodo-Kato cohomology (which I don’t understand at all – something about log-crystalline cohomology of the special fiber). They are both still open, although many interesting special cases have been solved.

  10. Thanks Eric. Is it true that all compact totally disconnected spaces with the same cardinality and no isolated points are homeomorphic?

    I guess I only thought of \mathbb{Z}_2 because the homeomorphism can be made bi-Lipschitz with a good normalization.

  11. Regarding Teichmüller lifts, there’s a nice geometric way to see what they are when the residue field is finite: The group G = mu_{p-1} is etale over Z_p (because its order is invertible). Therefore, by completeness of Z_p and the insensitivity of the etale topology to nilpotents, the reduction map G(Z_p) -> G(Z/p) is an isomorphism and, consequently, Z_p has (p-1)-st roots of unity which project down isomorphically to (Z/p)^* (i.e: basically Hensel’s lemma works).

  12. Is it true that all compact totally disconnected spaces with the same cardinality and no isolated points are homeomorphic?

    No. If you add in Hausdorff and second countable, then the answer is yes. One counter example to your question can be made by gluing two cantor sets together homeomorphically at every point but one. This space is like the real line with the origin doubled, but instead of the real line you have the cantor set. This space is not Hausdorff hence not homeomorphic to the cantor set.

  13. Also uncountible Fort space is another counter example. It is Hausdorff, compact, totally disconnected and contains no isolated points, but it is not even first countable so cannot be homeomorphic to the Cantor set.

    This is how it is defined: Take a countably infinite set X with a particular element p. We make X a topological space by declaring the open sets to be any subsets of X which have finite complement in X or for which the complement includes p. That’s it!

  14. No, in a fort space every point except p is isolated. Indeed, it’s just the one-point compactification of a discrete space.

    Here’s an example of two non-isomorphic totally disconnected compact Hausdorff spaces without isolated points (necessarily quite large and fairly intractible since they cannot be second countable): \beta N-N and {0,1}^R, where \beta N is the Stone-Cech compactification of a countable discrete space and {0,1}^R is the product of c two-point spaces. Both spaces have cardinality 2^c, but the second has a countable descending sequence of open sets whose intersection has empty interior while the first does not.

    More generally, the category of totally disconnected compact Hausdorff spaces (Stone spaces, for short) is dual to the category of Boolean algebras, with Stone spaces without isolated points corresponding to atomless Boolean algebras. The fact that the Cantor set is the only second-countable Stone space without isolated points corresponds to the fact that there is only one countable atomless Boolean algebra.

  15. Threaded comments could be nice.

    I take back my “counter example”. The Fort space does have (lots of) isolated points since any point other then p is open.

    There should be some counter example of this type though.

  16. For some reason Eric’s posts keep getting put in the spam folder, hence 22 comes after 20 in terms of when they were written, but before 20 in terms of which appeared on the site first.

  17. That’s wierd.

    Anyway, thanks Eric for pointing out my totally obvious mistake, even if I caught it before I saw your post. Interesting point about the Boolean algebras.

    I think I have another example to redeem myself. Basically copy the idea of the extended long line, replacing intervals [a, b) with Cantor sets.

    You take the first uncountable ordinal w and between each smaller ordinal a and it’s successor a+1 you add a copy of the Cantor set (minus an end point). You also add the ordinal w and give the whole thing the order topology.

    If you do this with interval [a, a+1) you get a compact space called the extended long line, so this should also be compact. It’s Hausdorff and not first-countable. It is also pretty easy to check that it is totally disconnected but without isolated points since it is locally just the Cantor set except at w, where you can check directly.

    If this works then it doesn’t seem too terrible a space, at least if you’re ok with the long line.

  18. Carnahan, I’m pretty sure that Grothendieck also proved an abstract version of the local monodromy theorem. I think it applies to any l-adic Galois representation over a local field with some conditions on the residue field, like not too many roots of unity or something.

    Also, I prefer to think of F_p as being a (big) circle and C((t)) as being a punctured infinitesimal disk. I’m pretty sure that if you have a family of varieties parameterized by the circle, then the eigenvalues of monodromy don’t have quasi-unipotence restrictions on them, just like for F_p. It would be nicer if F_p was more point-like than C((t)), but it’s not really so bad that the analogy isn’t perfect.

  19. Yeah, that example works. Another way of describing that space is as the one-point compactification of the union of aleph_1 copies of the Cantor set.

  20. James, I think you’re right. Another reason for \mathbb{F}_p do give a big circle comes from the philosophy that the ramification filtration on the Galois group of a local field reveals pregressively finer structure, and the unramified part is the first to go.

    I thought it should be a small circle, because I couldn’t think of anything that could go in the middle. That seems to be a bad reason, given the existence of objects like cycles on manifolds, etc.

  21. Anybody has a reference for the mentioned result that the algebraic closure of the field of p-adic numbers is isomorphic to the field of complex numbers, please?

  22. There’s a famous theorem that up to isomorphism, there is only one algebraically closed field of given transcendence degree over a prime field. The p-adics have a continuum’s worth of algebraically independent elements, so by this theorem its algebraic closure is isomorphic to C. I severely doubt this could be proven constructively though.

    This is a standard result in model theory (see e.g. Wilfrid Hodges, A Shorter Model Theory, chapter 8), and I’m surprised I couldn’t locate it in a standard algebra text (e.g. Lang or Jacobson), but maybe I didn’t try hard enough.

  23. Thanks, Todd. It’s good to know it’s true, but I don’t think I understand it better now than before. That’s perhaps why I have been looking at algebra textbooks myself, hoping to find hints “for the working mathematician;” still unsuccessful.

  24. notwendig, the direct result you want is an old theorem originally due to E. Steinitz, and says that two uncountable algebraically closed fields of the same characteristic and cardinality are isomorphic. It’s almost the same statement as the one I mentioned, whose proof is sketched here.

    The original source is

    E. Steinitz, Algebraischen Theorie der Körper, J. Reine Angew. Math. 137 (1910), 167-309.

    I’m not a model theorist, but from what I gather the model theorists think of the classification of algebraically closed fields (say over Q) according to transcendence degree along lines very similar to the classification of vector spaces (again say over Q) according to dimension. In particular, a common vocabulary involving terms like “spans’, ‘dependent’, ‘closure’, ‘dimension’, ‘Steinitz exchange lemma’ applies in parallel to both cases and to other suitably nice theories. It’s called the theory of minimal sets. Someday I’d like to learn it myself.

  25. Thanks again, Todd. That is exactly what I was looking for. These are more familiar terms now, and Steinitz’ article is fun to read – if you know some German. A bit more abstract, but still closer to mathematics than to logic, the theory of ‘matroids’ seems to capture the notions of ‘dependence’, ‘basis’, and ‘span’. Aigner’s ‘Combinatorial Theory’ seems to be a place to start for that.

  26. Where can I find info on the p-adic expansion of elements in C_p? What I mean is the generalization of the Laurent series in p for Q_p.

  27. Wishcow,

    I do not know of a characterization of elements in C_p as series in powers of p with Teichmüller coefficients. The condition that the order type be at most \omega^\omega is necessary, but Bjorn’s example of a sequence of nested balls with empty intersection shows that it is not sufficient.

  28. Wow.. wonderful.. I just stumped on this blog.. I need such discussions =) … the comment of Eric on August 23, 2007 (sorry I’m a little late.. I just saw this blog! -_-)

    “The fact that the Cantor set is the only second-countable Stone space without isolated points…”

    Is this “fact” easy to see or show? or do you have a reference where I can see it. Im trying to develop some easy examples of Stone Spaces that are first countable, I’m sure there is a work somewhere that already has this.

  29. hmm… regarding the first countability of Stone spaces.. the alexandroff compactification (1-point) of the natural numbers (with discrete topology) is probably not first countable.. the infinite point is not a point of first-countability…

  30. ah oh.. yes .. how totally forgetful of me. I did remember having seen that thing about Cantor Space. Thanks.

  31. Hi Scott,
    your article is really good. Do you have any idea about:
    if p and q are different primes then what can we say about
    Z_p (= p-adic integers ) and Z_q( = q-adic integers).. are they isomorphic as rings? are they homeomorphic as topological spaces?
    I think there cannot be ring isomorphism because, residue fields
    are of different sizes.

  32. They are not isomorphic as rings (or even groups), because multiplication by p has a nontrivial cokernel on Z_p, but is surjective on Z_q. They are homeomorphic as topological spaces, as Eric mentioned in comment 12, above.

  33. R. Khedkar,

    You are correct that Z_p and Z_q are not isomorphic as rings because they have different residue field characteristics. Indeed, they are not even isomorphic as abelian groups, because Z_p is not p-divisible, and Z_q is.

    However, they are homeomorphic (although not in a way that does anything nice to the algebraic structure) because they are both Cantor Spaces (see the statement of Brouwer’s theorem in that link).

  34. there is one remark in the above article that, Galois groups of these fields (i.e. Q_p) are pro-solvable.
    what is Galois group of C_p over Q_p?
    is it also pro-solvable?

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