Extensions of absolute values let pbe a prime and let q pbe an algebraic closure of q. Hensels lemma to show the existence of such a root z, we will first use. Hensels lemma tells us there is a unique 3adic integer such that 2 7 and 1 mod 3. Hensel s lemma concerning roots of polynomials over zp is introduced along with limits, concluding with a discussion of newton polygons.
If fx 2z px and some a2z p satis es jfaj p 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. Theorem hensel s 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. I where r is a semilocal ring with n invertible in r. The reader is referred to the very interesting paper of roquette 9 regarding the history of hensel s lemma and valuation theory in general. Euler, raphson, newton, puiseux, riemann, hurwitz, hensel april 20, 2015 divisionwithremainder of f0y by a linear in y polynomial y aproduces a remainder equal to evaluation of f0a.
S897 algebra and computation lecture 10 1 hensels lifting. Let a be an azumaya algebra of constant rank n over a hensel pair r. By the chinese remainder theorem, it suffices to solve the two. Additional notes on polynomial gcds, hensel construction. Hensel s lifting says we can obtain this if g and h are \relatively prime, and that the g and h we obtain are essentially unique. Let us try to convince you of the relevance of padic representation here. Hensel s lemma tells us when the root of a polynomial mod p can be lifted to a root in xp e g we saw that f x x 5 has no root in 72. An approach to hensels lemma trinity college dublin. Introduction hensel s lemma in z px is the following result about re ning an approximate solution of fx 0 to an exact solution. If fx 2z px and some a2z p satis es jfaj p of hense ls lemma over the padic integers cpp 19, january 1415, 2019, cascais, portugal 0 00 000 100 200 10 010 110 210 20 020 120 220 1 01 001 101 201 11 011 111 211 21 021 121 221 2 02 002 102 202 12 012 112 212 22 022 122 222 figure2. We are now ready to prove hensels lemma, which allows us to lift. Hensel s snake ditaxodon taeniatus, a snake endemic to southern brazil.
Thespecial case of one multiple factor becomes newton iteration for computing. A solution fx 1 0mod p satisfying f0x 1 6 0mod p can be lifted to a solution fx k 0mod pk. His father was sebastian hensel who was a landowner at the time that kurt was born, but later moved to berlin becoming a director of a construction firm. The proof of hensels lemma recursively constructs the solution. Reversing the order of the terms and noting that f0 fan and f1 f an w. The lifting method outlined in the proof is reminiscent of newtons.
Therefore hensels lemma with initial approximation. Formulated by kurt hensel, it predicts the existence of roots to a polynomial in the ring of padic integers given an initial approximated solution modulo prime p. Pdf new light on hensels lemma find and share research. A formal proof of hensels lemma over the padic integers arxiv. Conversely, if ais a quadratic nonresidue, by theorem 2. We will sketch in class a proof that there is a unique nonarchimedean absolute value jj p.
In this thesis, we employ elementary methods, namely hensels lemma and the chinese remainder theorem, to allow us to create three new infinite families of intersective. Hensels lemma now let us talk a little bit about the structure of padic. Hensels lemma daniel murfet april 8, 2005 let a be a ring which is complete for its a topology, where a is an ideal. Recall the notion of padic number representation as the representation of integers or reals in various number systems. The lemma is useful for finding and classifying solutions of polynomial equations modulo powers of primes with a minimum of computational difficulty. An element x2q p is a square if and only if it can be written x p2ny2 with n2z and y2z p a padic unit. General version 7 acknowledgments 11 references 12 1. The theorem can be thought of as saying, that a solution over mod pk can.
Multivariate hensel s lemma for complete rings yifei zhao contents 1. The idea is to use hensels lemma to solve this type of congruence. Sk1 of azumaya algebras over hensel pairs roozbeh hazrat abstract. In mathematics, hensels lemma, also known as hensels lifting lemma, named after kurt. 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. Polynomial congruences modulo pn and hensels lemma. Proof of hensel s lemma and the implicit function theorem7 references 8 in this set of notes, we prove that a complete ring satis es the multivariate hensel s lemma theorem 1. This might not seem like progress, but we will now show that under suitable conditions, once we have a root a. Hensel s lemma eiki norizuki in classical number theory, hensel lemma goes by this form. Hensel phelps construction, one of the largest general contractors and construction managers in the united states. Hensel s lemma i supposefxto be a polynomial indvariables with coef.
We show how certain relations occurring in the ring aa i. An approach to hensels lemma gary mcguire abstract. If p qthen the above argument yields roots of x2 r 0 mod qk for all kbecause ris a square mod qand we can still apply hensel s lemma. Was there a step in the proof in neukirch which did not assume discreteness that you did not understand. Hensel s 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 hensel s lifting lemma. Hensel s lemma daniel murfet april 8, 2005 let a be a ring which is complete for its a topology, where a is an ideal. Julius hensel was not only a student of agricultural chemistry, but also biochemistry and nutrition, and he related all these sciences, which 5. Proof of hensels lemma and the implicit function theorem. And since z p is an extension of z, the factorization of fx in z px is a re. The factorization to be lifted in this lemma is allowed to have multiple factors. Hensel lemma for nonsingular solutions although there is no analogue of lagranges theorem for prime power moduli, there is an algorithm for determining when a solution modulo p generates solutions to higher power moduli. One application is in factoring polynomials over z. By hensel s lemma these factors are the irreducible factors of fx in z px, reduced moduolo pz px.
The newton hensel lemma in this section we present another version of the hensel lemma. Lemma 1 hensel s lifting if r is a ring, i r is an ideal, and there exist f. Using newton polygons, a simple proof of a general hensel s lemma for separable polynomials over henselian fields is. Introduction the most familiar version of hensel s lemma states that if f is a polynomial. As mentioned earlier, usually we start with n 1 and k 0 in hensel s lemma.
Yiduan zheng reu 2020 edited university of chicago. The historical development of hensel s lemma is briefly discussed section 1. The completeness of a is essential for this purpose. 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. Let a be a complete local ring and fx 2 ax a monic polynomial. Then kis a local eld if and only if every equivalently, any closed ball in kis compact. Moreover, as p6 2, hensel s lemma implies the existence of a root of the appropriate quadratic modulo pk for all kand therefore a solution of px x2 qx2 rx2 qr 0 mod pk. Hensels lemma requires the valuation to be discrete mathematics. This paper shows how to weaken the hypotheses for hensel s lemma and some related theorems. Polynomial congruences and hensels lemma trinity university. This diagram displays initial segments of elements of z 3. Introduction the main goal of this paper is to introduce hensel s lemma. Pick c 2k with jcj 1 this is possible because jjis nontrivial.
Hensels original lemma showed how an approximative root can be refined to a root of. This means that it is the zero set of a set of polynomials with coef. The lifting method outlined in the proof is reminiscent of newtons method for solving equations. An intersective polynomial is a polynomial with integer coefficients that has no rational roots, but has a root modulo every integer greater than 1. Bread from stones he united into a composite science of life, which he called makrobiology. Henselian ring also hensel ring, a local ring in which hensel s lemma.
Hensels lemma eiki norizuki in classical number theory, hensel. Namely, one hopes that for some ythe modi ed value x. We first state the rp version and then the more general version hensel slemmaversion1 if f x c rp x and a ctlp satisfies f a o modp f a 0 modp thenthere is a unique bctlp st f b o and a b modp. Finally we study hensel lemma and see how it works with the general. Hensels lemma daniel katz department of mathematics. Hensel s lemma kurt hensel s 1897 interest in the padic numbers was for systematic solution of problems such as x2 1 mod 5 nfor all powers 5 of 5. Ribenboim 8 showed the logical equivalence between these and other hensel s lemmas. We now record a number of applications of hensel s lemma. We have f0 0 mod 2 and f1 0 mod 2, while f00 1 mod 2 and f01 0 mod 2. New light on hensels lemma connecting repositories. In this short article we present another approach to hensels lemma. The presence of x 1 rather than x n in the previous equality means that using hensel s lemma to generate a sequence fx ngrequires only one inversion modulo p.
The rst provides the promised result about lifting roots from k to a. Hensel s lemma for padics lemma hensel s lemma let f2q px and f0be the formal derivative. We will prove by induction that for each n 1 there is an a n2z psuch that fa n 0 mod pn, a n amod p. State and prove a generalization of the sophisticated form of hensel s lemma based on newtons iteration. We use the taylor expansion of f around r to write. In the present paper, a new proof of hensel s lemma. Chapter 9 hensels lemma university of saskatchewan. Euler, raphson, newton, puiseux, riemann, hurwitz, hensel. These polynomials have been difficult to find using traditional methods. New light on hensels lemma university of saskatchewan. Then the reduced whitehead group sk1a coincides with its reduction sk1aia.
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