Main points from lecture 8 the method and proof of hensels lemma 9 the finite from aa 1. Hensels lemma, described by gouvea as the most important algebraic property of the padic numbers, shows the existence of roots of polynomials over. The standard hypotheses for hensels lemma in several variables are slightly stronger than necessary, in the case that the jacobian determinant is not a unit. Suppose given a factorisation of fx modulo pis into several factors. The result called hensels lemma is a generalisation of a result due to kurt hensel on solving polynomial equations in padic number rings. 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.
To study diophantine equations over the integers, one often looks for local solutions, that is, solutions over the padic integers. Ito integrals theorem existence and uniqueness of ito integral suppose that v t 2m2 satis es the following. Therefore, using lemma 2, we have the following conclusion. For the necessity of the condition, when d 0 the proof is trivial. Assume that sis a partially ordered set, where every chain has an upper bound. Under certain assumptions we lift it to a true factorisation of fx in rx. Generalizations of hensels lemma and the nearest root. The historical development of hensels lemma is brie. Factorization if n is composite, how do we factor in polylog n time. Hensels l emma is a result that stipulates conditions for roots of polynomials modulo powers of primes to be lifted to roots modulo higher powers. Hensels lemma concerning roots of polynomials over zp is introduced along with limits, concluding with a discussion of newton polygons. In this paper, we extend the theorem of ore regarding factorization of polynomials over padic numbers to henselian valued fields of arbitrary rank thereby generalizing the main results of khanduja and kumar j pure appl algebra 216. Generalized hensels lemma article pdf available in proceedings of the edinburgh mathematical society 4203. We show how certain relations occurring in the ring aa i.
Chapter 9 hensels lemma university of saskatchewan. New and improved algorithms for computation in several fundamental polynomial operations are presented. Hensels lemma and its various modifications, such as, for instance, the henselrychlik theorem, are important tools for investigating problems of existence of roots of polynomials in valued fields. 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. According to the hausdor maximum principle, there exists a maximal chain c s. For polynomials over algebraically closed, valued fields, best possible results on continuity of roots section 4 and continuity of factors section 6 are demonstrated. It characterizes the meaning of a word by giving all the properties and only those properties that must be true. Hensels lemma daniel murfet april 8, 2005 let a be a ring which is complete for its a topology, where a is an ideal. Therefore, whenever i state a result let us restrict attention to results about univariate polynomials, to fix ideas as hensels lemma, i feel honorbound to inquire as to whether this result holds in a nonarchimedean normed field if and only if the field is henselian, i. The completeness of a is essential for this purpose. We would like to show you a description here but the site wont allow us. Using this and a general krasners lemma section 7, we give a short proof of a general hensel s lemma and show that it is, in a certain sense, best possible section 8.
Using newton polygons, a simple proof of a general hensels lemma for separable polynomials over henselian. Math texts, online classes, and more for students in grades 512. Were are looking at a field, k, complete with respect to a normalized nonarchimedean valuation, and let a be its discrete valuation ring all elements of k with absolute value less than or equal to 1 with maximal ideal mp, its residue field kam. New light on hensels lemma mathematics and statistics. May 21, 2016 for the love of physics walter lewin may 16, 2011 duration. Let f be monic in ax and f its image in kx, if fgh. Hensels lemma john stalker 1 composing polynomials in addition to adding and multiplying polynomials we can also \compose them. We will interpret z p as the projective limit of the rings zpnz, and. Let f be monic in ax and f its image in kx, if fgh in kx with g,h monic and. Theory of numbers lecture 6 primality, factoring, rsa. A formal proof of hensels lemma over the padic integers. We can now lift the extracondition in proposition 0. Singlefactor hensel lifting and its application to the.
It applies to certain complete topological rings, and now local rings that satisfy the conclusion of the lemma really. As an application, we derive the analogue of dedekinds theorem regarding splitting of. Reformulation of hensels lemma and extension of a theorem. This question is much simpler because hensels lemma often. Kurt and pauls paternal grandparents were painter wilhelm hensel and composer fanny mendelssohn. Thesimplestcase i shall look in this section and the next at the case when the variety is a hypersurface f 0, generically nonsingular, which is to say over qp.
The lemma is useful for finding and classifying solutions of polynomial equations modulo powers of primes with a minimum of computational difficulty. Aug 18, 2017 to study diophantine equations over the integers, one often looks for local solutions, that is, solutions over the padic integers. The lemma is useful for finding and classifying solutions of polynomial equations modulo powers of primes with a minimum of. In many cases, a lemma derives its importance from the theorem it aims to prove, however, a lemma can also turn out. Hensels lemma is a result that stipulates conditions for roots of polynomials modulo powers of primes to be lifted to roots modulo higher powers.
We will prove by induction that for each n 1 there is an a n 2z p such that fa n 0 mod pn, a n amod p. For that reason, it is also known as a helping theorem or an auxiliary theorem. Let r be a complete discrete valuation ring with maximal ideal generated by pi. Fanny was the sister of felix mendelssohn bartholdy, daughter of. This generalises the henselrychlik lemma, which covers the case of two factors. In many cases, a lemma derives its importance from the theorem it aims to prove, however, a lemma can also turn out to be more.
But, by lemma 4, tn, which leads to a contradiction. Hensels lemma now let us talk a little bit about the structure of padic. View notes hensels lemma notes from math 3461 at university of lethbridge. Jul 20, 2010 were are looking at a field, k, complete with respect to a normalized nonarchimedean valuation, and let a be its discrete valuation ring all elements of k with absolute value less than or equal to 1 with maximal ideal mp, its residue field kam. 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. Primality, factoring, rsa, hensels lemma, lecture 6 notes. To complete the proof of zorns lemma, it is enough to show that x has a maximal element. Introduction the most familiar version of hensels lemma states that if f is a polynomial. Main points from lecture 8 the method and proof of hensels. 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.
When n 2, we rst need a lemma pertaining to binary quadratic forms. Using this and a general krasners lemma section 7, we give a short proof of a general hensels lemma and show that it is, in a certain sense, best possible section 8. This question is much simpler because hensel s lemma often. For the love of physics walter lewin may 16, 2011 duration. Hensel s lemma tells us there is a unique 3adic integer such that 2 7 and 1 mod 3. In mathematics, informal logic and argument mapping, a lemma plural lemmas or lemmata is a generally minor, proven proposition which is used as a stepping stone to a larger result. This generalises the hensel rychlik lemma, which covers the case of two factors.
The quotation marks are there because polynomials are not functions. Hensels lemma understanding its proof physics forums. Lecture 7 5 some concluding observations may be of assistance. The theorem can be proved for the padics with significantly weaker. The disquisitiones arithmeticae latin for arithmetical investigations is a textbook of number theory written in latin 1 by carl friedrich gauss in when gauss was 21 and first published in when he was it is notable for having a revolutionary impact on the field of number theory as it not only turned the field truly rigorous and systematic but also paved the. Suppose there exists an integer x 0 such that px 0 0 mod pe and p0x 0 6 0 mod p. The lifting method outlined in the proof is reminiscent of newtons method for solving equations. Singlefactor hensel lifting and its application to the straightline complexity of certain polynomials. Home conferences stoc proceedings stoc 87 singlefactor hensel lifting and its application to the straightline complexity of certain polynomials. Therefore, whenever i state a result let us restrict attention to results about univariate polynomials, to fix ideas as hensel s lemma, i feel honorbound to inquire as to whether this result holds in a nonarchimedean normed field if and only if the field is henselian, i.