Higher Reciprocity Laws
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Algebraic Number Theory - Wikipedia, the Free Encyclopedia Page 1 of 5
Algebraic number theory - Wikipedia, the free encyclopedia Page 1 of 5 Algebraic number theory From Wikipedia, the free encyclopedia Algebraic number theory is a major branch of number theory which studies algebraic structures related to algebraic integers. This is generally accomplished by considering a ring of algebraic integers O in an algebraic number field K/Q, and studying their algebraic properties such as factorization, the behaviour of ideals, and field extensions. In this setting, the familiar features of the integers—such as unique factorization—need not hold. The virtue of the primary machinery employed—Galois theory, group cohomology, group representations, and L-functions—is that it allows one to deal with new phenomena and yet partially recover the behaviour of the usual integers. Contents 1 Basic notions 1.1 Unique factorization and the ideal class group 1.2 Factoring prime ideals in extensions 1.3 Primes and places 1.4 Units 1.5 Local fields 2 Major results 2.1 Finiteness of the class group 2.2 Dirichlet's unit theorem 2.3 Artin reciprocity 2.4 Class number formula 3 Notes 4 References 4.1 Introductory texts 4.2 Intermediate texts 4.3 Graduate level accounts 4.4 Specific references 5 See also Basic notions Unique factorization and the ideal class group One of the first properties of Z that can fail in the ring of integers O of an algebraic number field K is that of the unique factorization of integers into prime numbers. The prime numbers in Z are generalized to irreducible elements in O, and though the unique factorization of elements of O into irreducible elements may hold in some cases (such as for the Gaussian integers Z[i]), it may also fail, as in the case of Z[√-5] where The ideal class group of O is a measure of how much unique factorization of elements fails; in particular, the ideal class group is trivial if, and only if, O is a unique factorization domain . -
Number Theoretic Symbols in K-Theory and Motivic Homotopy Theory
Number Theoretic Symbols in K-theory and Motivic Homotopy Theory Håkon Kolderup Master’s Thesis, Spring 2016 Abstract We start out by reviewing the theory of symbols over number fields, emphasizing how this notion relates to classical reciprocity lawsp and algebraic pK-theory. Then we compute the second algebraic K-group of the fields pQ( −1) and Q( −3) based on Tate’s technique for K2(Q), and relate the result for Q( −1) to the law of biquadratic reciprocity. We then move into the realm of motivic homotopy theory, aiming to explain how symbols in number theory and relations in K-theory and Witt theory can be described as certain operations in stable motivic homotopy theory. We discuss Hu and Kriz’ proof of the fact that the Steinberg relation holds in the ring π∗α1 of stable motivic homotopy groups of the sphere spectrum 1. Based on this result, Morel identified the ring π∗α1 as MW the Milnor-Witt K-theory K∗ (F ) of the ground field F . Our last aim is to compute this ring in a few basic examples. i Contents Introduction iii 1 Results from Algebraic Number Theory 1 1.1 Reciprocity laws . 1 1.2 Preliminary results on quadratic fields . 4 1.3 The Gaussian integers . 6 1.3.1 Local structure . 8 1.4 The Eisenstein integers . 9 1.5 Class field theory . 11 1.5.1 On the higher unit groups . 12 1.5.2 Frobenius . 13 1.5.3 Local and global class field theory . 14 1.6 Symbols over number fields . -
Local-Global Methods in Algebraic Number Theory
LOCAL-GLOBAL METHODS IN ALGEBRAIC NUMBER THEORY ZACHARY KIRSCHE Abstract. This paper seeks to develop the key ideas behind some local-global methods in algebraic number theory. To this end, we first develop the theory of local fields associated to an algebraic number field. We then describe the Hilbert reciprocity law and show how it can be used to develop a proof of the classical Hasse-Minkowski theorem about quadratic forms over algebraic number fields. We also discuss the ramification theory of places and develop the theory of quaternion algebras to show how local-global methods can also be applied in this case. Contents 1. Local fields 1 1.1. Absolute values and completions 2 1.2. Classifying absolute values 3 1.3. Global fields 4 2. The p-adic numbers 5 2.1. The Chevalley-Warning theorem 5 2.2. The p-adic integers 6 2.3. Hensel's lemma 7 3. The Hasse-Minkowski theorem 8 3.1. The Hilbert symbol 8 3.2. The Hasse-Minkowski theorem 9 3.3. Applications and further results 9 4. Other local-global principles 10 4.1. The ramification theory of places 10 4.2. Quaternion algebras 12 Acknowledgments 13 References 13 1. Local fields In this section, we will develop the theory of local fields. We will first introduce local fields in the special case of algebraic number fields. This special case will be the main focus of the remainder of the paper, though at the end of this section we will include some remarks about more general global fields and connections to algebraic geometry. -
Cubic and Biquadratic Reciprocity
Rational (!) Cubic and Biquadratic Reciprocity Paul Pollack 2005 Ross Summer Mathematics Program It is ordinary rational arithmetic which attracts the ordinary man ... G.H. Hardy, An Introduction to the Theory of Numbers, Bulletin of the AMS 35, 1929 1 Quadratic Reciprocity Law (Gauss). If p and q are distinct odd primes, then q p 1 q 1 p = ( 1) −2 −2 . p − q We also have the supplementary laws: 1 (p 1)/2 − = ( 1) − , p − 2 (p2 1)/8 and = ( 1) − . p − These laws enable us to completely character- ize the primes p for which a given prime q is a square. Question: Can we characterize the primes p for which a given prime q is a cube? a fourth power? We will focus on cubes in this talk. 2 QR in Action: From the supplementary law we know that 2 is a square modulo an odd prime p if and only if p 1 (mod 8). ≡± Or take q = 11. We have 11 = p for p 1 p 11 ≡ (mod 4), and 11 = p for p 1 (mod 4). p − 11 6≡ So solve the system of congruences p 1 (mod 4), p (mod 11). ≡ ≡ OR p 1 (mod 4), p (mod 11). ≡− 6≡ Computing which nonzero elements mod p are squares and nonsquares, we find that 11 is a square modulo a prime p = 2, 11 if and only if 6 p 1, 5, 7, 9, 19, 25, 35, 37, 39, 43 (mod 44). ≡ q Observe that the p with p = 1 are exactly the primes in certain arithmetic progressions. -
Appendices A. Quadratic Reciprocity Via Gauss Sums
1 Appendices We collect some results that might be covered in a first course in algebraic number theory. A. Quadratic Reciprocity Via Gauss Sums A1. Introduction In this appendix, p is an odd prime unless otherwise specified. A quadratic equation 2 modulo p looks like ax + bx + c =0inFp. Multiplying by 4a, we have 2 2ax + b ≡ b2 − 4ac mod p Thus in studying quadratic equations mod p, it suffices to consider equations of the form x2 ≡ a mod p. If p|a we have the uninteresting equation x2 ≡ 0, hence x ≡ 0, mod p. Thus assume that p does not divide a. A2. Definition The Legendre symbol a χ(a)= p is given by 1ifa(p−1)/2 ≡ 1modp χ(a)= −1ifa(p−1)/2 ≡−1modp. If b = a(p−1)/2 then b2 = ap−1 ≡ 1modp,sob ≡±1modp and χ is well-defined. Thus χ(a) ≡ a(p−1)/2 mod p. A3. Theorem a The Legendre symbol ( p ) is 1 if and only if a is a quadratic residue (from now on abbre- viated QR) mod p. Proof.Ifa ≡ x2 mod p then a(p−1)/2 ≡ xp−1 ≡ 1modp. (Note that if p divides x then p divides a, a contradiction.) Conversely, suppose a(p−1)/2 ≡ 1modp.Ifg is a primitive root mod p, then a ≡ gr mod p for some r. Therefore a(p−1)/2 ≡ gr(p−1)/2 ≡ 1modp, so p − 1 divides r(p − 1)/2, hence r/2 is an integer. But then (gr/2)2 = gr ≡ a mod p, and a isaQRmodp. -
Sums of Gauss, Jacobi, and Jacobsthai* One of the Primary
JOURNAL OF NUMBER THEORY 11, 349-398 (1979) Sums of Gauss, Jacobi, and JacobsthaI* BRUCE C. BERNDT Department of Mathematics, University of Illinois, Urbana, Illinois 61801 AND RONALD J. EVANS Department of Mathematics, University of California at San Diego, L.a Jolla, California 92093 Received February 2, 1979 DEDICATED TO PROFESSOR S. CHOWLA ON THE OCCASION OF HIS 70TH BIRTHDAY 1. INTRODUCTION One of the primary motivations for this work is the desire to evaluate, for certain natural numbers k, the Gauss sum D-l G, = c e277inkl~, T&=0 wherep is a prime withp = 1 (mod k). The evaluation of G, was first achieved by Gauss. The sums Gk for k = 3,4, 5, and 6 have also been studied. It is known that G, is a root of a certain irreducible cubic polynomial. Except for a sign ambiguity, the value of G4 is known. See Hasse’s text [24, pp. 478-4941 for a detailed treatment of G, and G, , and a brief account of G, . For an account of G, , see a paper of E. Lehmer [29]. In Section 3, we shall determine G, (up to two sign ambiguities). Using our formula for G, , the second author [18] has recently evaluated G,, (up to four sign ambiguities). We shall also evaluate G, , G,, , and Gz4 in terms of G, . For completeness, we include in Sections 3.1 and 3.2 short proofs of known results on G, and G 4 ; these results will be used frequently in the sequel. (We do not discuss G, , since elaborate computations are involved, and G, is not needed in the sequel.) While evaluations of G, are of interest in number theory, they also have * This paper was originally accepted for publication in the Rocky Mountain Journal of Mathematics. -
Reciprocity Laws Through Formal Groups
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 141, Number 5, May 2013, Pages 1591–1596 S 0002-9939(2012)11632-6 Article electronically published on November 8, 2012 RECIPROCITY LAWS THROUGH FORMAL GROUPS OLEG DEMCHENKO AND ALEXANDER GUREVICH (Communicated by Matthew A. Papanikolas) Abstract. A relation between formal groups and reciprocity laws is studied following the approach initiated by Honda. Let ξ denote an mth primitive root of unity. For a character χ of order m, we define two one-dimensional formal groups over Z[ξ] and prove the existence of an integral homomorphism between them with linear coefficient equal to the Gauss sum of χ. This allows us to deduce a reciprocity formula for the mth residue symbol which, in particular, implies the cubic reciprocity law. Introduction In the pioneering work [H2], Honda related the quadratic reciprocity law to an isomorphism between certain formal groups. More precisely, he showed that the multiplicative formal group twisted by the Gauss sum of a quadratic character is strongly isomorphic to a formal group corresponding to the L-series attached to this character (the so-called L-series of Hecke type). From this result, Honda deduced a reciprocity formula which implies the quadratic reciprocity law. Moreover he explained that the idea of this proof comes from the fact that the Gauss sum generates a quadratic extension of Q, and hence, the twist of the multiplicative formal group corresponds to the L-series of this quadratic extension (the so-called L-series of Artin type). Proving the existence of the strong isomorphism, Honda, in fact, shows that these two L-series coincide, which gives the reciprocity law. -
Identifying the Matrix Ring: Algorithms for Quaternion Algebras And
IDENTIFYING THE MATRIX RING: ALGORITHMS FOR QUATERNION ALGEBRAS AND QUADRATIC FORMS JOHN VOIGHT Abstract. We discuss the relationship between quaternion algebras and qua- dratic forms with a focus on computational aspects. Our basic motivating problem is to determine if a given algebra of rank 4 over a commutative ring R embeds in the 2 × 2-matrix ring M2(R) and, if so, to compute such an embedding. We discuss many variants of this problem, including algorithmic recognition of quaternion algebras among algebras of rank 4, computation of the Hilbert symbol, and computation of maximal orders. Contents 1. Rings and algebras 3 Computable rings and algebras 3 Quaternion algebras 5 Modules over Dedekind domains 6 2. Standard involutions and degree 6 Degree 7 Standard involutions 7 Quaternion orders 8 Algorithmically identifying a standard involution 9 3. Algebras with a standard involution and quadratic forms 10 Quadratic forms over rings 10 Quadratic forms over DVRs 11 Identifying quaternion algebras 14 Identifying quaternion orders 15 4. Identifying the matrix ring 16 Zerodivisors 16 arXiv:1004.0994v2 [math.NT] 30 Apr 2012 Conics 17 5. Splitting fields and the Hilbert symbol 19 Hilbert symbol 19 Local Hilbert symbol 20 Representing local fields 22 Computing the local Hilbert symbol 22 6. The even local Hilbert symbol 23 Computing the Jacobi symbol 26 Relationship to conics 28 Date: May 28, 2018. 1991 Mathematics Subject Classification. Primary 11R52; Secondary 11E12. Key words and phrases. Quadratic forms, quaternion algebras, maximal orders, algorithms, matrix ring, number theory. 1 2 JOHN VOIGHT 7. Maximal orders 28 Computing maximal orders, generally 28 Discriminants 29 Computing maximal orders 29 Complexity analysis 32 Relationship to conics 34 8. -
Casting Retrospective Glances of David Hilbert in the History of Pure Mathematics
British Journal of Science 157 February 2012, Vol. 4 (1) Casting Retrospective Glances of David Hilbert in the History of Pure Mathematics W. Obeng-Denteh and S.K. Manu Department of Mathematics, Kwame Nkrumah University of Science and Technology, Kumasi, Ghana Abstract This study was conducted with the view of reconfirming the historical contributions of David Hilbert in the form of a review in the area of mathematics. It emerged that he contributed to the growth of especially the Nullstellensatz and was the first person to prove it. The implications and conclusions are that the current trend in the said area of mathematics could be appreciated owing to contributions and motivations of such a mathematician. Keywords: David Hilbert, Nullstellensatz, affine variety, natural number, general contribution 1.0 Introduction The term history came from the Greek word ἱστορία - historia, which means "inquiry, knowledge acquired by investigation‖( Santayana, …….) and it is the study of the human past. History can also mean the period of time after writing was invented. David Hilbert was a German mathematician who was born on 23rd January, 1862 and his demise occurred on 14th February, 1943. His parents were Otto and Maria Therese. He was born in the Province of Prussia, Wahlau and was the first of two children and only son. As far as the 19th and early 20th centuries are concerned he is deemed as one of the most influential and universal mathematicians. He reduced geometry to a series of axioms and contributed substantially to the establishment of the formalistic foundations of mathematics. His work in 1909 on integral equations led to 20th-century research in functional analysis. -
Primes and Quadratic Reciprocity
PRIMES AND QUADRATIC RECIPROCITY ANGELICA WONG Abstract. We discuss number theory with the ultimate goal of understanding quadratic reciprocity. We begin by discussing Fermat's Little Theorem, the Chinese Remainder Theorem, and Carmichael numbers. Then we define the Legendre symbol and prove Gauss's Lemma. Finally, using Gauss's Lemma we prove the Law of Quadratic Reciprocity. 1. Introduction Prime numbers are especially important for random number generators, making them useful in many algorithms. The Fermat Test uses Fermat's Little Theorem to test for primality. Although the test is not guaranteed to work, it is still a useful starting point because of its simplicity and efficiency. An integer is called a quadratic residue modulo p if it is congruent to a perfect square modulo p. The Legendre symbol, or quadratic character, tells us whether an integer is a quadratic residue or not modulo a prime p. The Legendre symbol has useful properties, such as multiplicativity, which can shorten many calculations. The Law of Quadratic Reciprocity tells us that for primes p and q, the quadratic character of p modulo q is the same as the quadratic character of q modulo p unless both p and q are of the form 4k + 3. In Section 2, we discuss interesting facts about primes and \fake" primes (pseu- doprimes and Carmichael numbers). First, we prove Fermat's Little Theorem, then show that there are infinitely many primes and infinitely many primes congruent to 1 modulo 4. We also present the Chinese Remainder Theorem and using both it and Fermat's Little Theorem, we give a necessary and sufficient condition for a number to be a Carmichael number. -
13 Lectures on Fermat's Last Theorem
I Paulo Ribenboim 13 Lectures on Fermat's Last Theorem Pierre de Fermat 1608-1665 Springer-Verlag New York Heidelberg Berlin Paulo Ribenboim Department of Mathematics and Statistics Jeffery Hall Queen's University Kingston Canada K7L 3N6 Hommage a AndrC Weil pour sa Leqon: goat, rigueur et pCnCtration. AMS Subiect Classifications (1980): 10-03, 12-03, 12Axx Library of Congress Cataloguing in Publication Data Ribenboim, Paulo. 13 lectures on Fermat's last theorem. Includes bibliographies and indexes. 1. Fermat's theorem. I. Title. QA244.R5 512'.74 79-14874 All rights reserved. No part of this book may be translated or reproduced in any form without written permission from Springer-Verlag. @ 1979 by Springer-Verlag New York Inc. Printed in the United States of America. 987654321 ISBN 0-387-90432-8 Springer-Verlag New York ISBN 3-540-90432-8 Springer-Verlag Berlin Heidelberg Preface Fermat's problem, also called Fermat's last theorem, has attracted the attention of mathematicians for more than three centuries. Many clever methods have been devised to attack the problem, and many beautiful theories have been created with the aim of proving the theorem. Yet, despite all the attempts, the question remains unanswered. The topic is presented in the form of lectures, where I survey the main lines of work on the problem. In the first two lectures, there is a very brief description of the early history, as well as a selection of a few of the more representative recent results. In the lectures which follow, I examine in suc- cession the main theories connected with the problem. -
Computing the Power Residue Symbol
Radboud University Master Thesis Computing the power residue symbol Koen de Boer supervised by dr. W. Bosma dr. H.W. Lenstra Jr. August 28, 2016 ii Foreword Introduction In this thesis, an algorithm is proposed to compute the power residue symbol α b m in arbitrary number rings containing a primitive m-th root of unity. The algorithm consists of three parts: principalization, reduction and evaluation, where the reduction part is optional. The evaluation part is a probabilistic algorithm of which the expected running time might be polynomially bounded by the input size, a presumption made plausible by prime density results from analytic number theory and timing experiments. The principalization part is also probabilistic, but it is not tested in this thesis. The reduction algorithm is deterministic, but might not be a polynomial- time algorithm in its present form. Despite the fact that this reduction part is apparently not effective, it speeds up the overall process significantly in practice, which is the reason why it is incorporated in the main algorithm. When I started writing this thesis, I only had the reduction algorithm; the two other parts, principalization and evaluation, were invented much later. This is the main reason why this thesis concentrates primarily on the reduction al- gorithm by covering subjects like lattices and lattice reduction. Results about the density of prime numbers and other topics from analytic number theory, on which the presumed effectiveness of the principalization and evaluation al- gorithm is based, are not as extensively treated as I would have liked to. Since, in the beginning, I only had the reduction algorithm, I tried hard to prove that its running time is polynomially bounded.