Dimension of an algebraic closure as a vector space over its base field. In this subsection, we always assume that a is a category with in. In mathematics, the padic number system for any prime number p extends the ordinary arithmetic of the rational numbers in a different way from the extension of the rational number system to the real and complex number systems. It is natural to seek a \smallest extension of k that is both complete and algebraically closed. An element of an extension of the field of rational numbers cf. Moreover, very interesting examples of padic galois representations arise from geometry. Here we will derive the structure of the padic complex numbers, that is, an algebraically closed, topologically complete field containing the p. Valuation rings, algebraic padic numbers algebraic padic numbers in the last section we defined the padic numbers as the completion of the rationals q, using the padic topology.
This is a survey on integral padic hodge theory, especially on the fontainela. A crucial difference is that the reals form an archimedean field, while the p padic numbers form a nonarchimedean field. Padic numbers 5 ostrowski proved that any eld complete with respect to an archimedean absolute value is isomorphic to r or c. As a consequence, any eld that can be endowed with an archimedean absolute value is isomorphic to a sub eld of c.
Koblitz, padic numbers, padic analysis, and zetafunctions, 2nd edition. When f is formally p adic but that there does not exist any proper algebraic formally p adic extension of f, then f is said to be p adically closed. A study of the padic frobenius lifts and padic periods. We describe a theory padic expansions for algebraic padic numbers in. We begin by emphasizing the properties of a nonarchimedean norm and exploring the topology it induces. The aim of the lecture was to discuss lowlevel algorithmics for padic numbers. The proof uses an iterative procedure starting with a polynomial ring in a very large number of variables and a maximal ideal of this ring that contains a certain proper. For people who prefer an algebraic approach, one can also define the p adic integers as an inverse limit of the rings z p n z and then define the p adic rational numbers as the fraction field of this inverse limit. Although he was unable to give complete proofs of the results announced in the complex and padic domains, the preprints, 14 played an important role. The eld c kis to be considered as an analogue of the complex numbers relative to k, and for k q pit is usually denoted c p. In particular, two padic numbers are considered to be close. Dec 06, 2012 such a visit offers a glimpse of a part of mathematics which is both important and fun, and which also is something of a meeting point between algebra and analysis. In this presentation we will discuss upto the algebraic closure of qp. A note on padic linear differential equations request pdf.
Hensel though they are foreshadowed in the work of his predecessor e. From the mathematical viewpoint r is not yet fully satisfying. In order to solve equations over the padic numbers, one. Let qpp be the algebraic closure of q, and let cp be its padic completion, which is an algebraically closed complete field with a valuation uniquely extended from qp and is also of. Jussieu 2009 81, 99177 c cambridge university press 99 doi. We let q denote the algebraic closure of q in c, and let z denote the integral closure of z in q. P adic numbers 5 ostrowski proved that any eld complete with respect to an archimedean absolute value is isomorphic to r or c. Over the last century, p adic numbers and p adic analysis have come to playa central role in modern number theory. In this case a continuous representation may have in. They are basic examples for things like valuations and pro nite groups. The padic numbers and finite field extensions of qp. For example, the domain could be the p adic integers z p, a profinite p group, or a p adic family of galois representations, and the image could be the p adic numbers q p or its algebraic closure.
This elementary introduction offers a broad understanding of p adic numbers. On the padic algebra and its applications 1269 nullstellensatz for the padic spectrum of a ring. Construction of cpand extension of padic valuations to c mike hamburg october 10, 2004 1 introduction here we will derive the structure of the padic complex numbers, that is, an algebraically closed, topologically complete. The source of a p adic lfunction tends to be one of two types. Thus, the completion of q with respect to the ordinary absolute value jjis r, and the algebraic closure of r is c, the.
We provide an introduction to the padic metric and its applications to algebra, analysis, topology, and number theory. The set of all p adic integers will be noted as z p whereas that of all p adic numbers will be noted as q p with p2p. However, as i would like to talk more about properties of the padic numbers, i will skip over this. In this note we will explain how padic numbers can be understood as defor. The padic numbers there are quite a few reasons to be interested in the padic numbers q p. It seems that hensels main motivation was the analogy between the ring of integers z, together with its. It is instructive to note that this allows to make zp algebraically definable. Macdu ees article titled the padic numbers of hensel from 1938 and h. For example, the field of p adic numbers is p adically closed, and so is the algebraic closure of the rationals inside it the field of p adic algebraic numbers. Introduction let kbe a eld complete with respect to a nontrivial nonarchimedean absolute value jj. An algebraic number is any complex number including real numbers that is a root of a nonzero polynomial that is, a value which causes the polynomial to equal 0 in one variable with rational coefficients or equivalently by clearing denominators with integer coefficients.
This paper will focus on the padic numbers and their properties. Eichlinghofen, the 28th august 2015 by gilles bellot tudortmunduniversity. They are useful for solving diophantine equations, using tools like hensels lemma and the hasse principle, which we wont say anything about. Current implementations of p adic numbers usually rely on so called zealous algorithms, which compute with truncated p adic expansions at a precision that can be specified by the user. An algebraic closure of a eld kis an algebraic extension lksuch that lis algebraically closed. Integral padic hodge theory and ramification of crystalline representations shin hattori abstract.
We nish this sketch of the p adic numbers by pointing out that the interaction between completion and algebraic closure is not as simple for the p adic numbers as for the real numbers. Madore revised 7th december 2000 in all that follows, pwill stand for a prime number. We can extend the padic norm even further, to the algebraic closure of qp. On the other hand, there is a much larger variety of elds with a nonarchimedean absolute value. Using zorns lemma or the weaker ultrafilter lemma, it can be shown that every field has an algebraic closure, and that the algebraic closure of a field k is unique up to an isomorphism that fixes every member of k. The reader is led into the rich structure of the fields qp and cp in a beautiful balance between analytic and algebraic aspects.
This is a wellwritten introduction to the world of p adic numbers. Construction of cp and extension of padic valuations to c. Blender game recording at the wrong time what to do with post. The padic completion of q and hensels lemma theodor christian herwig abstract. Notes on padic numbers and padically closed fields. Computations with padic numbers xavier caruso january 25, 2017 abstract this document contains the notes of a lecture i gave at the journ. Completing q with respect to p leads to the field qp of padic numbers. Consequently, authors often speak of the rather than an algebraic closure of k. Journal of number theory 23, 279284 1986 algebraic p adic expansions david lampert department of mathematics, university of michigan, ann arbor, michigan 48109 communicated by h. The extension is achieved by an alternative interpretation of the concept of closeness or absolute value. Can smartphones with the same camera sensor have different image quality. N, z, q, rand care the sets of respectively the natural numbers i.
R 0 with respect to which it is complete, and a continuous action of the galois group galq p in particular, galf. It took more than 25 years before p adic numbers were taken seriously by number theorists. The p adic numbers were motivated primarily by an attempt to bring the ideas and techniques of power series methods into number theory. Let q over bar be the algebraic closure of q in the field c p of p adic numbers. The p adic rational numbers are then the completion of the rational numbers with respect to this absolute value. Thurstons article titled the solution of padic equationsfrom 1943 published in the american mathematical monthly.