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 -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 is the one we learn in grade school, and for each prime , there is also a -adic metric , where is the largest such that if it exists, and if . The -adic metrics (unique up to normalization) are called ultrametrics, because they satisfy the following strengthening of the triangle inequality: . 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 , and completing yields the field of real numbers in the archimedean case, and the -adic fields for each prime . 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).
has a unique maximal compact subring, called , the ring of -adic integers, which is both the completion of under the -adic metric and the subset of elements of with non-negative valuation. in turn has a unique maximal ideal, , which generates the topology, and the quotient, called the residue field, is the field with elements. The topology on is fractal in nature, and in fact, is homeomorphic to the Cantor set. Elements of can be conveniently written as Laurent series in with coefficients in the residue field, but addition and multiplication are more complicated than in the Laurent series field because of nontrivial carries. You will typically see this presentation explained as writing rationals in base , but letting the expansion extend infinitely far to the left. Standard examples include and . We cannot let the expansion extend infinitely far in both directions, because multiplication becomes hard to define.
Now that we understand , I will spend the rest of the post discussing some distinguished extensions. We have the following strict inclusions:
The last three fields are ring-theoretically isomorphic to , 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 , a maximal tamely ramified extension, and , a maximal abelian extension, but they are beyond the scope of this post. I’ll add some references at the end.
is known as a maximal unramified extension of . It is given by adjoining all th roots of unity, for . The elements of this field can be represented by Laurent series in , with coefficients in , 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 , in other words, prime-to- roots of unity or zero. Alternatively, one can work with Witt vectors, representing the coefficients as . 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.”).
is an algebraic closure of . Typical elements not in are roots of , -power roots of unity, and more exotic examples like roots of (discovered by Chevalley). Their expansions in powers of do not necessarily have exponents that form a discrete subset of the rationals. Bjorn Poonen’s undergraduate thesis has partially expanded in powers of two, and the set of exponents with nonzero coefficients seems to have order type , 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.
is the -adic completion of . One can generate transcendental elements in by summing a rapidly decaying sequence of elements of increasing degree such as , 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 that doesn’t converge in . is one of the most commonly used base fields for -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.
is the spherical completion of , 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, is not spherically complete, and in fact, one can reasonably say that most nested sequences of balls in have empty intersection. One example (pointed out to me by Bjorn) is given by taking the balls of the form , 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 ? The answer quite simple, and is found in Poonen’s undergraduate thesis. The elements are exactly those power series with coefficients given by Teichmüller representatives of , 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 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 (e.g., the value group is -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 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 . 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 , which is not -adically complete, and its completion is not spherically complete, closely mirroring the local field case. The absolute Galois group of is , isomorphic to that of , but instead of a Frobenius, there is the monodromy operator . 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 as a three-ball with the primes forming embedded knots.
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 -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 , 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.