DOCSLIB.ORG

Appendices A. Quadratic Reciprocity Via Gauss Sums

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
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. ♣ 2 A4. Theorem The mapping a → χ(a) is a homomorphism from Fp to {±1}. Proof. We compute ab a b ≡ (ab)(p−1)/2 = a(p−1)/2b(p−1)/2 ≡ p p p so χ(ab)=χ(a)χ(b). ♣ A5. Theorem If g is a primitive root mod p, then χ(g)=−1, so g is a quadratic nonresidue (from now on abbreviated QNR) mod p. Consequently, exactly half of the integers 1,2,...,p − 1 are QR’s and half are QNR’s. Proof.Ifh2 ≡ g mod p then h is a primitive root with twice the period of g, which is impossible. Thus by (A4), χ(ag)=−χ(a), so a → ag gives a bijection between QR’s and QNR’s. ♣ A6. The First Supplementary Law From the definition (A2) and the fact that (p − 1)/2isevenifp ≡ 1 mod 4 and odd if p ≡ 3 mod 4, we have −1 1ifp ≡ 1mod4 =(−1)(p−1)/2 = p −1ifp ≡ 3mod4. A7. Definition Let K be a field of characteristic = p such that K contains the p-th roots of unity. Let ζ ∈ K be a primitive p-th root of unity. Define the Gauss sum by p−1 a τ = ζa. p p a=1 A8. Theorem 2 − (p−1)/2 τp =( 1) p. Proof. From the definition of Gauss sum and (A4) we have p−1 ab τ 2 = ζa+b. p p a,b=1 3 For each a, we can sum over all c such that b ≡ ac mod p. (As c ranges over 1,... ,p− 1, ac also takes all values 1,... ,p− 1.) Thus p−1 p−1 a2c τ 2 = ζa+ac. p p a=1 c=1 a2 Since p = 1, this simplifies to p−1 p−1 c τ 2 = ζa(1+c) . p p c=1 a=1 If 1 + c ≡ 0modp then 1,ζ1+c,ζ2(1+c),... ,ζ(p−1)(1+c) runs through all the roots (zeros) of Xp − 1 (note that ζp = 1). But the coefficient of Xp−1 is 0, so the sum of the roots is 0. Therefore the sum of ζ1+c,ζ2(1+c),... ,ζ(p−1)(1+c) is -1. If 1 + c ≡ 0modp, then we are summing p − 1 ones. Consequently, p−2 c −1 τ 2 = − +(p − 1) . p p p c=1 ≡ ↔ − − −1 (Note that 1 + c 0modp c = p 1.) We can sum from 1 to p 1ifweadd p , hence p−1 c −1 τ 2 = − + p . p p p c=1 The sum is 0 by (A5), and the result follows from (A6). ♣ A9. The Law of Quadratic Reciprocity Let p and q be odd primes, with p = q. Then p q =(−1)(p−1)(q−1)/4. q p Thus if either p or q is congruent to 1 mod 4, then p is a QR mod q if and only if q is a QR mod p; and if both p and q are congruent to 3 mod 4, then p isaQRmodq if and only if q is a QNR mod p. Proof. Let K have characteristic q and contain the p-th roots of unity. For example, take p K to be the splitting field of X − 1 over Fq. Then q 2 (q−1)/2 τp =(τp ) τp which by (A8) is (q−1)/2 (p−1)/2 (−1) p τp. 4 Thus q − (p−1)(q−1)/4 (q−1)/2 τp =( 1) p τp. But by the binomial expansion applied to the definition of τp in (A7), p−1 a τ q = ζaq p p a=1 (Recall that K has characteristic q,soaq = a in K.) Let c ≡ aq mod p and note that q−1 q = p p 1 because the product of the two terms is ( p ) = 1. Thus p−1 c q q τ q = ζc = τ . p p p p p c=1 q We now have two expressions (not involving summations) for τp ,so q (−1)(p−1)(q−1)/4 p(q−1)/2 = . p (q−1)/2 p F Since p =(q ) by (A2), the above equation holds not only in K but in q, hence p can be written as a congruence mod q. Finally, multiply both sides by ( q ) to complete the proof. ♣ A10. The Second Supplementary Law 2 2 =(−1)(p −1)/8 p so 2 1ifp ≡±1mod8 = p −1ifp ≡±3mod8. Thus if p ≡ 1 or 7 mod 8, then 2 is a QR mod p, and if p ≡ 3or5mod8,then2isa QNR mod p. Proof. Let K be a field of characteristic p containing the 8th roots of unity, and let ζ be a primitive 8th root of unity. Define τ = ζ + ζ−1. Then τ 2 = ζ2 + ζ−2 +2. Now ζ2 and ζ−2 are distinct 4th roots of unity, not ±1, so they must be negatives of each other (analogous to i and −i in C). Therefore τ 2 = 2. Modulo p we have 2 2 τ p =(τ 2)(p−1)/2 τ =2(p−1)/2 τ = τ = (ζ + ζ−1). p p 5 But by definition of τ, τ p =(ζ + ζ−1)p = ζp + ζ−p. Now (again as in C) ζp + ζ−p and ζ + ζ−1 will coincide if p ≡±1 mod 8, and will be negatives of each other if p ≡±3mod 8. In other words, 2 ζp + ζ−p =(−1)(p −1)/8(ζ + ζ−1). Equating the two expressions for τ p, we get the desired result. We can justify the appeal to the complex plane by requiring that K satisfy the constraints ζ2 + ζ−2 = 0 and 2 ζp + ζ−p =(−1)(p −1)/8(ζ + ζ−1). ♣ A11. Example We determine whether 113 is a QR mod 127: 113 127 ≡ ( 127 )=(113 ) because 113 1mod4; 127 14 a ( 113 )=(113 ) because ( p ) depends only on the residue class of a mod p; 14 2 7 7 ≡ ( 113 )=(113 )( 113 )=(113 ) by (A4) and the fact that 113 1mod8; 7 113 ≡ ( 113 )=( 7 ) because 113 1mod4; 113 1 ≡ ( 7 )=(7 ) because 113 1mod7 1 and since ( 7 ) = 1 by inspection, 113 is a QR mod 127. Explicitly, 113 +13(127) = 1764 = 42 2. A12. The Jacobi Symbol Let Q be an odd positive integer with prime factorization Q = q1 ···qs. The Jacobi symbol is defined by s a a = . Q q i=1 i It follows directly from the definition that a a a a a aa = , = . Q Q QQ Q Q Q Also, a a a ≡ a mod Q ⇒ = Q Q because if a ≡ a mod Q then a ≡ a mod qi for all i =1,... ,s. Quadratic reciprocity and the two supplementary laws can be extended to the Jacobi symbol, if we are careful. 6 A13. Theorem If Q is an odd positive integer then −1 =(−1)(Q−1)/2. Q Proof. We compute − s − s 1 1 (q −1)/2 s (q −1)/2 = = (−1) j =(−1) j=1 j . Q q j=1 j j=1 Now if a and b are odd then ab − 1 a − 1 b − 1 (a − 1)(b − 1) − + = ≡ 0mod2, 2 2 2 2 hence a − 1 b − 1 ab − 1 + ≡ mod 2. 2 2 2 We can apply this result repeatedly (starting with a = q1,b = q2) to get the desired formula. ♣ A14. Theorem If Q is an odd positive integer, then 2 2 =(−1)(Q −1)/8. Q Proof. As in (A13), s s 2 2 (q2−1)/8 = = (−1) j . Q q j=1 j j=1 But if a and b are odd then a2b2 − 1 a2 − 1 b2 − 1 (a2 − 1)(b2 − 1) − + = ≡ 0mod8. 8 8 8 8 a2−1 (a−1)(a+1) 2 − ≡ In fact 8 = 8 is an integer and b 1 0 mod 8. (Just plug in b =1, 3, 5, 7.) Thus a2 − 1 b2 − 1 a2b2 − 1 + ≡ mod 8 8 8 8 and we can apply this repeatedly to get the desired result.
Load more

Recommended publications
  • Quadratic Reciprocity and Computing Modular Square Roots.Pdf
    12 Quadratic reciprocity and computing modular square roots In §2.8, we initiated an investigation of quadratic residues. This chapter continues this investigation. Recall that an integer a is called a quadratic residue modulo a positive integer n if gcd(a, n) = 1 and a ≡ b2 (mod n) for some integer b. First, we derive the famous law of quadratic reciprocity. This law, while his- torically important for reasons of pure mathematical interest, also has important computational applications, including a fast algorithm for testing if an integer is a quadratic residue modulo a prime. Second, we investigate the problem of computing modular square roots: given a quadratic residue a modulo n, compute an integer b such that a ≡ b2 (mod n). As we will see, there are efficient probabilistic algorithms for this problem when n is prime, and more generally, when the factorization of n into primes is known. 12.1 The Legendre symbol For an odd prime p and an integer a with gcd(a, p) = 1, the Legendre symbol (a j p) is defined to be 1 if a is a quadratic residue modulo p, and −1 otherwise. For completeness, one defines (a j p) = 0 if p j a. The following theorem summarizes the essential properties of the Legendre symbol. Theorem 12.1. Let p be an odd prime, and let a, b 2 Z. Then we have: (i) (a j p) ≡ a(p−1)=2 (mod p); in particular, (−1 j p) = (−1)(p−1)=2; (ii) (a j p)(b j p) = (ab j p); (iii) a ≡ b (mod p) implies (a j p) = (b j p); 2 (iv) (2 j p) = (−1)(p −1)=8; p−1 q−1 (v) if q is an odd prime, then (p j q) = (−1) 2 2 (q j p).
    [Show full text]
  • QUADRATIC RECIPROCITY Abstract. the Goals of This Project Are to Have
    QUADRATIC RECIPROCITY JORDAN SCHETTLER Abstract. The goals of this project are to have the reader(s) gain an appreciation for the usefulness of Legendre symbols and ultimately recreate Eisenstein's slick proof of Gauss's Theorema Aureum of quadratic reciprocity. 1. Quadratic Residues and Legendre Symbols Definition 0.1. Let m; n 2 Z with (m; n) = 1 (recall: the gcd (m; n) is the nonnegative generator of the ideal mZ + nZ). Then m is called a quadratic residue mod n if m ≡ x2 (mod n) for some x 2 Z, and m is called a quadratic nonresidue mod n otherwise. Prove the following remark by considering the kernel and image of the map x 7! x2 on the group of units (Z=nZ)× = fm + nZ :(m; n) = 1g. Remark 1. For 2 < n 2 N the set fm+nZ : m is a quadratic residue mod ng is a subgroup of the group of units of order ≤ '(n)=2 where '(n) = #(Z=nZ)× is the Euler totient function. If n = p is an odd prime, then the order of this group is equal to '(p)=2 = (p − 1)=2, so the equivalence classes of all quadratic nonresidues form a coset of this group. Definition 1.1. Let p be an odd prime and let n 2 Z. The Legendre symbol (n=p) is defined as 8 1 if n is a quadratic residue mod p n < = −1 if n is a quadratic nonresidue mod p p : 0 if pjn: The law of quadratic reciprocity (the main theorem in this project) gives a precise relation- ship between the \reciprocal" Legendre symbols (p=q) and (q=p) where p; q are distinct odd primes.
    [Show full text]
  • Quadratic Reciprocity Laws*
    JOURNAL OF NUMBER THEORY 4, 78-97 (1972) Quadratic Reciprocity Laws* WINFRIED SCHARLAU Mathematical Institute, University of Miinster, 44 Miinster, Germany Communicated by H. Zassenhaus Received May 26, 1970 Quadratic reciprocity laws for the rationals and rational function fields are proved. An elementary proof for Hilbert’s reciprocity law is given. Hilbert’s reciprocity law is extended to certain algebraic function fields. This paper is concerned with reciprocity laws for quadratic forms over algebraic number fields and algebraic function fields in one variable. This is a very classical subject. The oldest and best known example of a theorem of this kind is, of course, the Gauss reciprocity law which, in Hilbert’s formulation, says the following: If (a, b) is a quaternion algebra over the field of rational numbers Q, then the number of prime spots of Q, where (a, b) does not split, is finite and even. This can be regarded as a theorem about quadratic forms because the quaternion algebras (a, b) correspond l-l to the quadratic forms (1, --a, -b, ub), and (a, b) is split if and only if (1, --a, -b, ub) is isotropic. This formulation of the Gauss reciprocity law suggests immediately generalizations in two different directions: (1) Replace the quaternion forms (1, --a, -b, ub) by arbitrary quadratic forms. (2) Replace Q by an algebraic number field or an algebraic function field in one variable (possibly with arbitrary constant field). While some results are known concerning (l), it seems that the situation has never been investigated thoroughly, not even for the case of the rational numbers.
    [Show full text]
  • Selmer Groups and Quadratic Reciprocity 3
    SELMER GROUPS AND QUADRATIC RECIPROCITY FRANZ LEMMERMEYER Abstract. In this article we study the 2-Selmer groups of number fields F as well as some related groups, and present connections to the quadratic reci- procity law in F . Let F be a number field; elements in F × that are ideal squares were called sin- gular numbers in the classical literature. They were studied in connection with explicit reciprocity laws, the construction of class fields, or the solution of em- bedding problems by mathematicians like Kummer, Hilbert, Furtw¨angler, Hecke, Takagi, Shafarevich and many others. Recently, the groups of singular numbers in F were christened Selmer groups by H. Cohen [4] because of an analogy with the Selmer groups in the theory of elliptic curves (look at the exact sequence (2.2) and recall that, under the analogy between number fields and elliptic curves, units correspond to rational points, and class groups to Tate-Shafarevich groups). In this article we will present the theory of 2-Selmer groups in modern language, and give direct proofs based on class field theory. Most of the results given here can be found in 61ff of Hecke’s book [11]; they had been obtained by Hilbert and Furtw¨angler in the§§ roundabout way typical for early class field theory, and were used for proving explicit reciprocity laws. Hecke, on the other hand, first proved (a large part of) the quadratic reciprocity law in number fields using his generalized Gauss sums (see [3] and [24]), and then derived the existence of quadratic class fields (which essentially is just the calculation of the order of a certain Selmer group) from the reciprocity law.
    [Show full text]
  • 4 Patterns in Prime Numbers: the Quadratic Reciprocity Law
    4 Patterns in Prime Numbers: The Quadratic Reciprocity Law 4.1 Introduction The ancient Greek philosopher Empedocles (c. 495–c. 435 b.c.e.) postulated that all known substances are composed of four basic elements: air, earth, fire, and water. Leucippus (fifth century b.c.e.) thought that these four were indecomposable. And Aristotle (384–322 b.c.e.) introduced four properties that characterize, in various combinations, these four elements: for example, fire possessed dryness and heat. The properties of compound substances were aggregates of these. This classical Greek concept of an element was upheld for almost two thousand years. But by the end of the nineteenth century, 83 chem- ical elements were known to exist, and these formed the basic building blocks of more complex substances. The European chemists Dmitry I. Mendeleyev (1834–1907) and Julius L. Meyer (1830–1895) arranged the elements approx- imately in the order of increasing atomic weight (now known to be the order of increasing atomic numbers), which exhibited a periodic recurrence of their chemical properties [205, 248]. This pattern in properties became known as the periodic law of chemical elements, and the arrangement as the periodic table. The periodic table is now at the center of every introductory chemistry course, and was a major breakthrough into the laws governing elements, the basic building blocks of all chemical compounds in the universe (Exercise 4.1). The prime numbers can be considered numerical analogues of the chemical elements. Recall that these are the numbers that are multiplicatively indecom- posable, i.e., divisible only by 1 and by themselves.
    [Show full text]
  • Notes on Mod P Arithmetic, Group Theory and Cryptography Chapter One of an Invitation to Modern Number Theory
    Notes on Mod p Arithmetic, Group Theory and Cryptography Chapter One of An Invitation to Modern Number Theory Steven J. Miller and Ramin Takloo-Bighash September 5, 2006 Contents 1 Mod p Arithmetic, Group Theory and Cryptography 1 1.1 Cryptography . 1 1.2 Efficient Algorithms . 3 1.2.1 Exponentiation . 3 1.2.2 Polynomial Evaluation (Horner’s Algorithm) . 4 1.2.3 Euclidean Algorithm . 4 1.2.4 Newton’s Method and Combinatorics . 6 1.3 Clock Arithmetic: Arithmetic Modulo n ................................ 10 1.4 Group Theory . 11 1.4.1 Definition . 12 1.4.2 Lagrange’s Theorem . 12 1.4.3 Fermat’s Little Theorem . 14 1.4.4 Structure of (Z=pZ)¤ ...................................... 15 1.5 RSA Revisited . 15 1.6 Eisenstein’s Proof of Quadratic Reciprocity . 17 1.6.1 Legendre Symbol . 17 1.6.2 The Proof of Quadratic Reciprocity . 18 1.6.3 Preliminaries . 19 1.6.4 Counting Lattice Points . 21 1 Abstract We introduce enough group theory and number theory to analyze in detail certain problems in cryptology. In the course of our investigations we comment on the importance of finding efficient algorithms for real world applica- tions. The notes below are from An Invitation to Modern Number Theory, published by Princeton University Press in 2006. For more on the book, see http://www.math.princeton.edu/mathlab/book/index.html The notes below are Chapter One of the book; as such, there are often references to other parts of the book. These references will look something like ?? in the text. If you want additional information on any of these references, please let me know.
    [Show full text]
  • 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.
    [Show full text]
  • The Legendre Symbol and Its Properties
    The Legendre Symbol and Its Properties Ryan C. Daileda Trinity University Number Theory Daileda The Legendre Symbol Introduction Today we will begin moving toward the Law of Quadratic Reciprocity, which gives an explicit relationship between the congruences x2 ≡ q (mod p) and x2 ≡ p (mod q) for distinct odd primes p, q. Our main tool will be the Legendre symbol, which is essentially the indicator function of the quadratic residues of p. We will relate the Legendre symbol to indices and Euler’s criterion, and prove Gauss’ Lemma, which reduces the computation of the Legendre symbol to a counting problem. Along the way we will prove the Supplementary Quadratic Reciprocity Laws which concern the congruences x2 ≡−1 (mod p) and x2 ≡ 2 (mod p). Daileda The Legendre Symbol The Legendre Symbol Recall. Given an odd prime p and an integer a with p ∤ a, we say a is a quadratic residue of p iff the congruence x2 ≡ a (mod p) has a solution. Definition Let p be an odd prime. For a ∈ Z with p ∤ a we define the Legendre symbol to be a 1 if a is a quadratic residue of p, = p − 1 otherwise. ( a Remark. It is customary to define =0 if p|a. p Daileda The Legendre Symbol Let p be an odd prime. Notice that if a ≡ b (mod p), then the congruence x2 ≡ a (mod p) has a solution if and only if x2 ≡ b (mod p) does. And p|a if and only if p|b. a b Thus = whenever a ≡ b (mod p).
    [Show full text]
  • Arxiv:1408.0235V7 [Math.NT] 21 Oct 2016 Quadratic Residues and Non
    Quadratic Residues and Non-Residues: Selected Topics Steve Wright Department of Mathematics and Statistics Oakland University Rochester, Michigan 48309 U.S.A. e-mail: [email protected] arXiv:1408.0235v7 [math.NT] 21 Oct 2016 For Linda i Contents Preface vii Chapter 1. Introduction: Solving the General Quadratic Congruence Modulo a Prime 1 1. Linear and Quadratic Congruences 1 2. The Disquisitiones Arithmeticae 4 3. Notation, Terminology, and Some Useful Elementary Number Theory 6 Chapter 2. Basic Facts 9 1. The Legendre Symbol, Euler’s Criterion, and other Important Things 9 2. The Basic Problem and the Fundamental Problem for a Prime 13 3. Gauss’ Lemma and the Fundamental Problem for the Prime 2 15 Chapter 3. Gauss’ Theorema Aureum:theLawofQuadraticReciprocity 19 1. What is a reciprocity law? 20 2. The Law of Quadratic Reciprocity 23 3. Some History 26 4. Proofs of the Law of Quadratic Reciprocity 30 5. A Proof of Quadratic Reciprocity via Gauss’ Lemma 31 6. Another Proof of Quadratic Reciprocity via Gauss’ Lemma 35 7. A Proof of Quadratic Reciprocity via Gauss Sums: Introduction 36 8. Algebraic Number Theory 37 9. Proof of Quadratic Reciprocity via Gauss Sums: Conclusion 44 10. A Proof of Quadratic Reciprocity via Ideal Theory: Introduction 50 11. The Structure of Ideals in a Quadratic Number Field 50 12. Proof of Quadratic Reciprocity via Ideal Theory: Conclusion 57 13. A Proof of Quadratic Reciprocity via Galois Theory 65 Chapter 4. Four Interesting Applications of Quadratic Reciprocity 71 1. Solution of the Fundamental Problem for Odd Primes 72 2. Solution of the Basic Problem 75 3.
    [Show full text]
  • Arxiv:0804.2233V4 [Math.NT]
    October 26, 2018 MEAN VALUES WITH CUBIC CHARACTERS STEPHAN BAIER AND MATTHEW P. YOUNG Abstract. We investigate various mean value problems involving order three primitive Dirichlet characters. In particular, we obtain an asymptotic formula for the first moment of central values of the Dirichlet L-functions associated to this family, with a power saving in the error term. We also obtain a large-sieve type result for order three (and six) Dirichlet characters. 1. Introduction and Main results Dirichlet characters of a given order appear naturally in many applications in number theory. The quadratic characters have seen a lot of attention due to attractive questions to ranks of elliptic curves, class numbers, etc., yet the cubic characters have been relatively neglected. In this article we are interested in mean values of L-functions twisted by characters of order three, and also large sieve-type inequalities for these characters. Our first result on such L-functions is the following Theorem 1.1. Let w : (0, ) R be a smooth, compactly supported function. Then ∞ → q (1) ∗ L( 1 , χ)w = cQw(0) + O(Q37/38+ε), 2 Q (q,3)=1 χ (mod q) X χX3=χ 0 b where c> 0 is a constant that can be given explicitly in terms of an Euler product (see (23) below), and w is the Fourier transform of w. Here the on the sum over χ restricts the sum ∗ to primitive characters, and χ0 denotes the principal character. This resultb is most similar (in terms of method of proof) to the main result of [L], who considered the analogous mean value but for the case of cubic Hecke L-functions on Q(ω), ω = e2πi/3.
    [Show full text]
  • By Leo Goldmakher 1.1. Legendre Symbol. Efficient Algorithms for Solving Quadratic Equations Have Been Known for Several Millenn
    1. LEGENDRE,JACOBI, AND KRONECKER SYMBOLS by Leo Goldmakher 1.1. Legendre symbol. Efficient algorithms for solving quadratic equations have been known for several millennia. However, the classical methods only apply to quadratic equations over C; efficiently solving quadratic equations over a finite field is a much harder problem. For a typical in- teger a and an odd prime p, it’s not even obvious a priori whether the congruence x2 ≡ a (mod p) has any solutions, much less what they are. By Fermat’s Little Theorem and some thought, it can be seen that a(p−1)=2 ≡ −1 (mod p) if and only if a is not a perfect square in the finite field Fp = Z=pZ; otherwise, it is ≡ 1 (or 0, in the trivial case a ≡ 0). This provides a simple com- putational method of distinguishing squares from nonsquares in Fp, and is the beginning of the Miller-Rabin primality test. Motivated by this observation, Legendre introduced the following notation: 8 0 if p j a a <> = 1 if x2 ≡ a (mod p) has a nonzero solution p :>−1 if x2 ≡ a (mod p) has no solutions. a (p−1)=2 a Note from above that p ≡ a (mod p). The Legendre symbol p enjoys several nice properties. Viewed as a function of a (for fixed p), it is a Dirichlet character (mod p), i.e. it is completely multiplicative and periodic with period p. Moreover, it satisfies a duality property: for any odd primes p and q, p q (1) = hp; qi q p where hm; ni = 1 unless both m and n are ≡ 3 (mod 4), in which case hm; ni = −1.
    [Show full text]
  • Quadratic Reciprocity Via Linear Algebra
    QUADRATIC RECIPROCITY VIA LINEAR ALGEBRA M. RAM MURTY Abstract. We adapt a method of Schur to determine the sign in the quadratic Gauss sum and derive from this, the law of quadratic reciprocity. 1. Introduction Let p and q be odd primes, with p = q. We define the Legendre symbol (p/q) to be 1 if p is a square modulo q and 16 otherwise. The law of quadratic reciprocity, first proved by Gauss in 1801, states− that (p/q)(q/p) = ( 1)(p−1)(q−1)/4. − It reveals the amazing fact that the solvability of the congruence x2 p(mod q) is equivalent to the solvability of x2 q(mod p). ≡ There are many proofs of this law≡ in the literature. Gauss himself published five in his lifetime. But all of them hinge on properties of certain trigonometric sums, now called Gauss sums, and this usually requires substantial background on the part of the reader. We present below a proof, essentially due to Schur [2] that uses only basic notions from linear algebra. We say “essentially” because Schur’s goal was to deduce the sign in the quadratic Gauss sum by studying the matrix A = (ζrs) 0 r n 1, 0 s n 1 ≤ ≤ − ≤ ≤ − where ζ = e2πi/n. For the case of n = p a prime, Schur’s proof is reproduced in [1] and a ‘slicker’ proof was later supplied by Waterhouse [3]. Our point is to show that Schur’s proof can be modified to determine tr A when n is an odd number and this allows us to deduce the law of quadratic reciprocity.
    [Show full text]