Hensel's lemma for p-adic numbers pdf

Hensels lemma, described by gouva as the most important algebraic property of the padic numbers, shows the existence of roots of polynomials over p provided an initial seed point. But the latter has to be mitself by the previous lemma. Browse other questions tagged abstractalgebra number theory padic number theory or ask your own question. As mentioned earlier, usually we start with n 1 and k 0 in hensels lemma. Thus, we cannot use the trivial case of hensels lemma. As you might guess, there is a version of hensels lemma which works for any prime, and not just 3. Well need to use a lemma that is a simple corollary of sperners lemma. Hensels lemma kurt hensels 1897 interest in the padic numbers was for systematic solution of problems such as x2 1 mod 5 nfor all powers 5 of 5. Hensel s lemma, also known as hensel s lifting lemma, named after kurt hensel, is a result in modular arithmetic, stating that if a polynomial equation has a simple root modulo a prime number p, then this root corresponds to a unique root of the same equation modulo any higher power of p, which can be found by iteratively lifting the solution modulo successive powers of p. Chapter 1 of number theory by borevich and shafarevich proves the following result.

For a prime p 1 mod 4, every p adic integer is a sum of two squares of p adic integers. Hensel s lifting lemma is the criterion for the existence of the root of p adic function f on the set of p adic integers. Consider a quadratic form with rational coe cients, fx 1x d x i j a ijx ix j. Padics, hensels lemma and strassmans theorem yuchen chen abstract.

The field of padic numbers p and the ring of padic integers p are essential constructions of modern number theory. Suppose, for one of the three numbers k 0,1,2, that pk 0 inz3 andp. The padic numbers are most simply a field extension of q, the rational numbers. Dirichlets theorem on primes in arithmetic progression.

In this paper we consider the problem of finding the roots of padic functions. The most important thing to keep in mind is that numbers with small padic absolute values are. Newtons method for padic numbers jeff january 25, 20 at 8. The theorem can be proved for the padics with significantly weaker hypotheses than for general rings. Notes for a talk at the ams regional meeting in providence, ri, october, please note that. A new class of padic lipschitz functions and hensels lemma 3. We conclude with a thorough analysis of the padic aspects of the bernoulli numbers. A formal proof of hensels lemma over the padic integers. The main tool for nding the roots of padic functions that map the ring of padic integers into itself, is a classical result hensels lifting lemma. We let c p be the p adic completion of q p, called the eld of p adic complex numbers. Hensels lemma 10 acknowledgments references introduction the padics present a particularly rich subject.

Theorem hensels lifting lemma for padic case let fx 2z px be a polynomial with integer padic coe cients and f0x 2z px be its formal derivative. Readers who have seen the construction of the real numbers. The proof lies at the intersection of algebraic and analytic reasoning and demonstrates how the lean mathematical library handles such a heterogeneous topic. In mathematics the padic number system for any prime number p extends the ordinary arithmetic. Speci cally, we draw upon how hensels lemma allows us to determine the existence of rational solutions to a homogenous polynomial of degree 2 in 3 variables. Hensels lemma, described by gouvea as the most important algebraic property of the padic numbers, shows the existence of roots of polynomials over. The padic numbers are less well known than the others, but they play a fundamental role in number theory and in other parts of mathematics. A sequence of integer numbers is a cauchy sequence if and only if their di.

Explicit methods for solving diophantine equations. Then we expand hensels lemma to apply to a system of multivariable polynomials, and. Hensel codes of square roots of padic numbers article pdf available in applicable analysis and discrete mathematics 41 april 2010 with 266 reads how we measure reads. We will adopt a more general approach, due to kurschak 19, which is based on absolute values. Hensel s lifting lemma is characterized by the following circumstances. These exotic numbers or so they appeared at first are now wellestablished in the mathematical world and used more and more by physicists as well. New light on hensels lemma university of saskatchewan.

Hensels lemma, described by gouvea as the most important algebraic. Turning information modulo one power of pinto similar information modulo a higher power of pcan be interpreted as improving an approximation. Thus, the solutions constructed by the method of hensels lemma form a cauchy 2. The p adic integers z p correspond to the case k 0 0, and invertible elements in this ring have k 0 0. A quantitative version of hensels lemma mathoverflow. In particular, padic numbers are considered to be close when. Hensels lemma now let us talk a little bit about the structure of padic. Hensels lemma posted on november 7, 20 by matt baker tagged diffiehellman key exchange discrete logarithms hensels lemma padic analysis personal anecdotes primitive roots splitting fields commentsno comments on primitive roots, discrete logarithms, and padic numbers primitive roots, discrete logarithms, and padic numbers. Slide 11 the padics as a foundational tool hensel showed, in several papers and in his books, that one could. Lewis vrije universiteit amsterdam the netherlands r. Then, we end our tour of q p with a proof of hensel s lemma a result describing a general approach to building p adic numbers. Hensels lemma concerning roots of polynomials over zp is introduced along with limits.

This book offers a selfcontained presentation of basic padic. The padic numbers were first developed by kurt hensel in 1897 and for each prime p they can be considered as a completion of the rational numbers. This criterion allows us to determine whether f has a root in z p by solving a finite number of congruences. A new class of padic lipschitz functions and hensels lemma 3 we. The function may seem contrived and quite abstract, but it is worth taking time to understand the nature of this function well, as it will become central to our discussion of the size of padic numbers. Then we expand hensels lemma to apply to a system of multivariable polynomials, and show that fermats last theorem has padic solutions. Hensel, and this invention led to substantial developments in number theory, where padic numbers are now as natural as ordinary real numbers. So we see that the notion of an archimedean absolute value has its origin in archimedes lemma which states that for all nonzero rationals. We study padic continuous functions in several variables taking values on zp. We begin by noting that the p in padic stands for a prime number. An introduction to the padic numbers semantic scholar. Newest henselslemma questions mathematics stack exchange. 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.

If p 1 mod 4 then a theorem of fermat from number theory says 1 mod pis a square. Bleher noticed in 19 that the set of purely fractional padic numbers is an example of hierarchical lattice. The goal of this project is to develop a background in the study of hensels padic numbers, including a discussion of nonarchimedean valuations, completion. Recall that each x2q pcan be expressed in the form 1. To my eyes, the most glaring omission was a book by fernando gouvea, originally published in, entitled padic numbers. Kurt hensel 18611941 discovered the padic numbers around the turn of the century. The extension is achieved by an alternative interpretation of the concept of closeness or absolute value. Ill give a proof of hensels lemma at the end of these notes. However we can still apply a modified version by going one power of two up. Hensel though they are foreshadowed in the work of his predecessor e.

For example, taking p to be 3, we may talk about the 3adic numbers. Hensels lifting lemma the main tool for nding the roots of padic functions that map the ring of padic integers into itself, is a classical result hensels lifting lemma. This is an expository paper on an introduction to padic numbers. The padic absolute values on q are very di erent from the archimedean absolute value. Hensels lemma is closely analogous to the newtonraphson method. This elementary introduction offers a broad understanding of padic numbers. Theorem hensels lifting lemma for padic case let fx 2z px be a polynomial with integer padic coe cients and f0x. Specifically, we draw upon how hensels lemma allows us to determine the. The presence of x 1 rather than x n in the previous equality means that using hensels lemma to generate a sequence fx ngrequires only one inversion modulo p. Corrections, comments, and suggestions are welcome.

Please note that this is still a preliminary report. The analysis includes the p adic numbers q pand the ring of integers z p. Introduction in the padic integers, congruences are approximations. We present an example of solving a quadratic congruence modulo a composite using hensels lemma and the chinese remainder theorem. About a century later, padic numbers was one of the most crucial topics of pure mathematics to investigate. In the case, where the function is defined by a polynomial with integer padic coefficients, using hensels lifting lemma helps us find the roots of the padic function we generalize hensels lifting lemma for a wider class of padic functions, namely, the functions which satisfy the lipschitz condition with. The ring of padic integers zp is the completion of the ring of integers. We will interpret z p as the projective limit of the rings zpnz, and. The lifting method outlined in the proof is reminiscent of newtons method for solving equations. Then f has a nonzero root in q dif and only if f has a nonzero. Introduction padic numbers were first described by german mathematician kurt hensel. Hensels lemma is a result that stipulates conditions for roots of polynomials modulo powers of primes to be lifted to roots modulo higher powers. For example, jxj p 1 for in nitely many rational numbers xall those whose numerator and denominator are prime to p, whereas jxj 1 only for the two values 1. Number theory solving quadratic congruences with hensel.

The lemma is useful for finding and classifying solutions of polynomial equations modulo powers of primes with a minimum of computational difficulty. A formal proof of hensels lemma over the padic integers arxiv. We also prove a hensels lifting lemma for this class of. Color the outer three vertices a, b, c as 1, 2, and 3 respectively. Hensels lemma, valuations, and padic numbers jordan bell jordan. The theorem can be proved for the padics with significantly weaker. Hensel simply defined padic integers by their power series expansions. A formal proof of hensels lemma over the padic integers robert y. If you want to avoid the analysis but still learn and write about the padics, you can explore their interesting topology. We will start by constructing qp by completing the rational numbers with respect to the padic norm.

In particular, if and turn out to be natural numbers, then their sum as a padic integer is no different from their sum as a natural number. 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. This is followed\ud by a rigorous construction of the padic numbers, and we show that padic numbers are complete. Relaxed padic hensel lifting for algebraic systems. Pdf p adic numbers an introduction uploaded by nora roberts, introduction this section is an informal introduction to p adic numbers using examples from the ring. Turning information modulo one power of pinto similar. In this section we summarize the basic aspects of the. It seems that hensels main motivation was the analogy between the ring of integers z, together with its. After inducting on n, this proof follows nearly automati cally using leans ring tactic. Pdf the historical development of hensels lemma is briefly discussed section 1.