In mathematics, 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. An important result in valuation theory which gives information on finding roots of polynomials. When undertaking a first course in number theory, students enjoy actively engaging with the properties and relationships of numbers. This is the simplest example i know of where a result in number theory can be proved with the banach fixed point theorem. A solid introduction to analytic number theory, including full proofs of dirichlets theorem and the prime number theorem. A formal proof of hensels lemma over the padic integers. Hensels lemma is closely analogous to the newtonraphson method. Algebraic number theory 20112012 math user home pages. Introduction to number theory 0366214001, 0366214002. Featured is a chapter on how to apply the theory of newton polygons to determine galois groups of polynomials over the rational number field. Number theory olympiad an olympiad level study of number theory involves familiarity with intermediate topics to a high level, a few new topics, and a highly developed proof writing ability.
Syllabus theory of numbers mathematics mit opencourseware. This is a book about prime numbers, congruences, secret messages, and elliptic curves that you can read cover to cover. 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. Multivariate hensels lemma for complete rings yifei zhao contents 1. Featured on meta planned maintenance scheduled for wednesday, february 5, 2020 for data explorer. Note that these problems are simple to state just because a topic is accessibile does not mean that it is easy. Browse other questions tagged padic number theory hensels lemma or ask your own question. On congruence properties of the partition function. Diophantine analysis mathematical association of america.
Literature by and about kurt hensel in the german national library catalogue. Let be a polynomial whose coefficients are in and suppose satisfies. New in this edition are coverage of padic numbers, hensel s lemma, multiple zetavalues, and elliptic curve methods in primality testing. It grew out of undergraduate courses that the author taught at harvard, uc san diego, and the university of washington. Number theory solving polynomial congruences with hensel. Online shopping from a great selection at books store. Newtons method for padic numbers jeff january 25, 20 at 8. The book opens by introducing material usually found in an undergraduate number theory book. The proof supplies a formula for computing s, but for now we can find s 12 by inspection.
Turning information modulo one power of pinto similar information modulo a higher power of pcan be interpreted as improving an approximation. The book treats the subject informally, making the text accessible to nonexperts. Gerhard rosenberger now in its second edition, this textbook provides an introduction and overview of number theory based on the density and properties of the prime numbers. If you want to avoid the analysis but still learn and write about the padics, you can explore their interesting topology. Number theory solving polynomial congruences with hensels lemma. For example, here are some problems in number theory that remain unsolved. Hensel s lemma is a result that stipulates conditions for roots of polynomials modulo powers of primes to be lifted to roots modulo higher powers.
See also the number of real quadratic fields having units of negative norm by peter stevenhagen. Although it is not useful computationally, it has theoretical significance, being involved in some proofs of quadratic reciprocity. See also modular forms notes from 20056 and 201011 and 2014. Number theory an introduction via the density of primes. We give an example of solving a polynomial congruence modulo a power of a prime. Introduction to number theory is dedicated to concrete questions about integers, to place an emphasis on problem solving by students. Number theory solving polynomial congruences with hensels. This is quite comprehensive and has a nice collection of topics and exercises. Number theory solving quadratic congruences with hensel s lemma duration. Browse other questions tagged elementary number theory modulararithmetic hensels lemma or ask your own question.
Consider two monic polynomials f and f of common degree n 1 with integral coefficients in a henselian field k, v. Topics covered include primes, congruences, quadratic reciprocity, diophantine equations, irrational numbers, continued fractions, and partitions. It applies to certain complete topological rings, and now local rings that satisfy the conclusion of the lemma really a theorem are called henselian rings. Which book you usedplease tell me the name of book. Using newton polygons, a simple proof of a general hensel s lemma for separable polynomials over henselian fields is.
All of the essential topics are covered, such as the fundamental theorem of arithmetic, theory of congruences, quadratic reciprocity, arithmetic functions, and the distribution of primes. See, for example, a new development of the theory of algebraic numbers, by g. Hensels lemma can be found in many elementary number theory text books. No one can predict when what seems to be a most obscure theorem may suddenly be called upon to play some vital and hitherto unsuspected role. Was hensels lemma originally used for proving some other theorem. Introduction to number theory essential textbooks in. The field of padic numbers p and the ring of padic integers p are essential constructions of modern number theory. We can now state and prove the promised general hensel s lemma. There is no formal distinction between a lemma and a theorem, only one of intention see theorem terminology. A bit expensive, but if you want to own one book on elementary number theory, this ones a pretty good candidate. Newest numbertheory questions history of science and. This lecture note is an elementary introduction to number theory with no algebraic prerequisites. The historical development of hensel s lemma is briefly discussed section 1. Recall that a prime number is an integer greater than 1 whose only positive factors are 1 and the number itself.
Hensel showed, in several papers and in his books, that one could give a simpler account of the theory of divisibility in algebraic number. Suppose a be a commutative ring that is complete with respect to ideal m. Proof of hensels lemma and the implicit function theorem7 references 8 in this set of notes, we prove that a complete ring satis es the multivariate hensels lemma theorem 1. In dworks book on gfunctions and kedlayas book on padic differential equations, a version of hensels lemma is proved by the banach fixed point theorem on a finitedimensional padic vector space of polynomials. This is used to obtain results on the divisibility of stirling numbers of the second kind which generalise results of davis. The novel method of finding of roots of padic functions is applicable in the class of. Intersective polynomials and hensels lemma ubc open collections. The result called hensels lemma is a generalisation of a result due to kurt hensel on solving polynomial equations in padic number rings. Let f 2zx be a polynomial with integer coef cients. Hensel s lemma, valuations, and padic numbers jordan bell jordan.
As motivation for further study, the final chapter introduces iwasawa theory. Hensels lemma, valuations, and padic numbers jordan bell jordan. Hensels lemma tells us there is a unique 3adic integer. In part ii we will discuss the basics of local fields. The systematic study of number theory was initiated around 300b. Introduction in the padic integers, congruences are approximations. The writing style here is, at times, lively and cute. Let there be given a factorisation f gh with monic g and h. Subsequent topics explore quadratic reciprocity, hensels lemma, padic. The lifting method outlined in the proof is reminiscent of newtons method for solving equations. A formal proof of hensels lemma over the padic integers arxiv. Introduction to number theory essential textbooks in mathematics.
However, a lemma can be considered a minor result whose sole purpose is to help prove a theorem a step in the direction of proof or a short theorem appearing at an intermediate stage in a proof. Any introductory book on number theory will be useful. Use hensel s lemma to solve congruences of the form fx. Gausss lemma in number theory gives a condition for an integer to be a quadratic residue. In this video i use the lifting the exponent lemma and the theory of orders to solve an imo 1999 number theory problem. Algebraic numbers, finite continued fractions, infinite continued fractions, periodic continued fractions, lagrange and pell, eulers totient function, quadratic residues and nonresidues, sums of squares and quadratic forms. Applications of banach fixed point theorem to number theory. In mathematics, hensels lemma, also known as hensels 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. Most of number theory has very few practical applications. If this is true, what reference book gives a proof of this fact stated as above. That does not reduce its importance, and if anything it enhances its fascination. Find materials for this course in the pages linked along the left. Number theory hensels lemma youtube we prove hensels lemma, which is related to finding solutions to polynomial congruences modulo powers of.
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