Quadratic reciprocity law

HANOI PEDAGOGICAL UNIVERSITY DEPARTMENT OF MATHEMATICS PHUONG THI ANH THU QUADRATIC RECIPROCITY LAW GRADUATION THESIS Major: Algebra Hanoi - 2019 HANOI PEDAGOGICAL UNIVERSITY DEPARTMENT OF MATHEMATICS PHUONG THI ANH THU QUADRATIC RECIPROCITY LAW GRADUATION THESIS Major: Algebra Supervisor: Dr Tran Nam Trung Hanoi - 2019 Thesis Acknowledgement I would like to express my gratitude to the teachers of the Department of Mathematics, Hanoi Pedagogical University 2, the teachers in the algebra group as well as the teachers involved The lecturers have imparted valuable knowledge and facilitated for me to complete the courses and the thesis In particular, I would like to express my deep respect and gratitude to Dr Tran Nam Trung, who directly guided me and helped me complete this thesis Due to the limit of time, capacity and conditions the thesis can not avoid errors Then, I look forward to receiving valuable comments from teachers and friends Hanoi, May - 2019 Student Phuong Thi Anh Thu Graduation thesis PHUONG THI ANH THU Thesis Assurances This thesis is completed after self-study and research, and the enthusiastic guidance of Dr Tran Nam Trung In this thesis, I refer to the relevant documents systematically in the references section I assure that this thesis does not coincide with any essay and I will take full responsibility for my pledge Hanoi, May - 2019 Student Phuong Thi Anh Thu i Contents Thesis Acknowledgements Thesis Assurance i Preface 1 Quadratic Residues 1.1 Congruences 1.2 Quadratic residues 1.3 Legendre Symbol Quadratic Reciprocity 11 2.1 Quadratic Reciprocity 11 2.2 Some applications 16 Conclusions 21 References 22 ii Graduation thesis PHUONG THI ANH THU Preface The purpose of this thesis is to discuss about congruences by means of a remarkable result of Gauss known as the Quadratic Reciprocity Law In the preceding thesis, the problem of solving such a congruence as x2 ≡ a (mod m) was reduced to the case of a prime modulo p The question remains as to whether x2 ≡ a (mod p) does or does not have solution This question can be narrowed to the case x2 ≡ q (mod p), where q is also a prime The Quadratic Reciprocity Law states that if p and q are distinct odd primes, the two congruences x2 ≡ q (mod p) and x2 ≡ p (mod q) are either both solvable or both not solvable Besides, to study the Quadratic congruence equation, Legendre gave a technical notation to consider the equation x2 ≡ a (mod p) does or does not havesolution And we call this notation as Legendre Symbol :    +1 if a is a quadratic residue and p a    a p = −1 if a is a quadratic non-residue modulo p     0 if p | a This a very useful shorthand for dealing with Quadratic residues and Quadratic non-residues Thesis is to study the Quadratic Reciprocity Law Let us describe the content of this thesis In Chapter 1, we will introduce Quadratic Residue modulo an inte1 Graduation thesis PHUONG THI ANH THU ger n The Quadratic residue of n are the integers which are squares modulo n We will particularly study quadratic residues modulo an odd prime p We will discuss Eulers criterion, which specifies when an integer is a quadratic residue modulo p Whether an integer a is a quadratic residue modulo p is indicated by a symbol called Legendres symbol and verifies its basic properties In Chapter 2, we will discuss about the Quadratic Reciprocity Law It is one of the most important theorems in an Elementary number theory This is theorem about modular arithmetic that gives conditions for the solvability of quadratic equations modulo prime numbers The law of Quadratic Reciprocity is not just an academic questions in number theory, it has far reaching consequences, for example in cryptography There are over a hundred proofs of the Quadratic Reciprocity Law In this chapter, we present one proof in Section 2.1 which is completely elementary and involves keeping track of integer points in intervals It is satisfying because one can understand every detail without much abstraction, but it is unsatisfying because it is difficult to conceptualize what is going on The main references are [1, 2, 3] Chapter Quadratic Residues In this chapter, we will review some result about Quadratic residue which will be used to in the next chapter We begin with congruences theory and its properties in Section 1.1 The definition of Quadratic residue and non-residue and some important properties will be present in Section 1.2 Section 1.3 defines the Legendre Symbol and verifies its basic properties, states Euler’s Criterion 1.1 Congruences Definition 1.1.1 Let m be a fixed positive integer and two integers a, b If m not zero, divides the difference a − b, we say that a is congruent to b modulo m and denoted by a ≡ b (mod m) Graduation thesis PHUONG THI ANH THU Otherwise, if a − b is not divisible by m, we say that a is not congruent to b modulo m, and in this case we denote by a ≡ b (mod m) Example 1.1.2 We see that: • ≡ (mod 3), (−2) ≡ (mod 5) because − = (−1) · and (−2) − = (−1) · • ≡ (mod 4), 11 ≡ (mod 3) because (2 − 1) and (11 − 1) • Any two integers are congruent modulo • If two integers either both even or both odd, then they are congruent modulo Remark 1.1.3 a ≡ b (mod m) if and only if a and b have the same remainder when divided by m The following theorem gives the basic properties of congruences: Theorem 1.1.4 Let m > be fixed and a, b, c, d be any integers Then, a ≡ b (mod m), b ≡ a (mod m), and (a − b) ≡ (mod m) are equivalent statements If a ≡ b (mod m) and b ≡ c (mod m) then a ≡ c (mod m) If a ≡ b (mod m) and c ≡ d (mod m) then (a + c) ≡ (b + d) (mod m) Graduation thesis PHUONG THI ANH THU If a ≡ b (mod m) and c ≡ d (mod m) then ac ≡ bd (mod m) If a ≡ b (mod m) and d | m, d > 0, then a ≡ b (mod d) If a ≡ b (mod m) then ac ≡ bc (mod mc) for c > If a ≡ b (mod m) then ak ≡ bk (mod m) for k > Example 1.1.5 Find the remainder when 20172019 divided by 10 Solution Since 2017 ≡ (mod 10), by Theorem 1.1.4 (7) we get 20172019 ≡ 72019 (mod 10) On the other hand, 74 ≡ (mod 10) Thus, 72016 = (74 )504 ≡ (mod 10) This implies that 72019 ≡ 73 ≡ (mod 10), so the remainder when 20172019 divided by 10 is Theorem 1.1.6 (Fermat’s little theorem) Let p be a prime number and a an integer Suppose that p a Then ap−1 ≡ (mod p) Proof We consider the first p − positive multiples of a; that is the integers a, 2a, 3a, · · · , (p − 1)a We claim that none of these numbers is congruent modulo p to any other, nor is any congruent to zero Graduation thesis PHUONG THI ANH THU Proof As c2 ≡ (p − c)2 (mod p) for c = 1, , p − 1, the number of quadratic residues is at most (p − 1)/2 On the other hand, for all i py and qx < py Observe that S1 = (x, y) ∈ Z2 | ≤ x ≤ (p − 1)/2, ≤ y < qx/p so that (p−1)/2 |S1 | = x=1 qx p Similarly, S2 = (x, y) ∈ Z2 | ≤ y ≤ (q − 1)/2, ≤ x < py/q so (q−1)/2 |S2 | = y=1 14 py q Graduation thesis PHUONG THI ANH THU Since |S| = |S1 | + |S2 |, we have (p−1)/2 j=1 jx + p (q−1)/2 j=1 pj (p − 1)(q − 1) = q The theorem now follows by Theorem 2.1.3 The quadratic reciprocity law together with basic properties of quadratic residues give us an efficient way to compute Legendre symbols −42 61 Example 2.1.5 Compute Solution Since −42 = (−1) · · · 7, we have −42 61 −1 61 61 61 61 = −1 61 61 61 61 = (−1)60/2 = = (−1)(61 −1)/8 = −1 = (−1)(2/2)(60/2) = (−1)(6/2)(60/2) = (−1)(4/2)(6/2) 61 = 61 = 7 = 5 =1 = (−1)(5 −1)/8 Thus, −42 61 = −1 61 61 15 61 61 = = −1 Graduation thesis 2.2 PHUONG THI ANH THU Some applications We now apply the theory of quadratic residues to show that a prime of the form 4n + can be written as a sum of two squares First note that (a + b)2 (c + d2 ) = (ac + db)2 + (ad − bc)2 (2.1) for any numbers a, b, c and d Theorem 2.2.1 Let p is an odd prime number Then, p is a sum of two squares if and only if p is of the form 4n + Proof If p is a sum of two squares, say p = x2 + y Since p is a prime, x and y are nonzero First x and y are coprime to p Assume on the contrary that p | x, then p | y since p = x2 + y , and then p | y Let x = pa and y = pb We have p = x2 + y = p2 (a2 + b2 ), so = p(a2 + b2 ) In particular, p | 1, a contradiction Since (x, p) = (y, p) = 1, and x2 = −y (mod p) We have 1= x2 p = −y p = y2 p −1 p = −1 p = (−1)(p−1)/2 , so p is of the form 4n + Conversely, assume that p is of the form 4n + Then, −1 p = (−1)(p−1)/2 = Thus, the congruence z + ≡ (mod p) has solution So there exists a positive integers z and m such that mp = z + 12 16 Graduation thesis PHUONG THI ANH THU Now, consider the more general congruence,x2 + y ≡ (mod p) Let x1 andx2 be the residues of x, y (mod p) such that |x1 |, |y1 | < p/2 Then, p2 2 < p < m = (x1 + y1 ) < · p p (2.2) Let m = m0 be the least positive integer for which mp is representable as a sum of two squares m0 p = x21 + y12 (2.3) By Inequality (2.2), m0 < p We will prove that m0 = 1, and the theorem follows Assume on the contrary that m0 > Let x2 , y2 be the residues of x1 , y1 (mod m0 ) such that |x2 |, |y2 | ≤ m0 Observe that x22 + y22 ≡ x21 + y12 ≡ (mod m0 ) This implies that there is r such that rm0 = x22 + y22 (2.4) We claim that r > Indeed, if r = 0, then x2 = y2 = =⇒ m0 | x1 , m0 | y1 =⇒ m20 | m0 p =⇒ m0 | p But < m0 < p, this contradicts that p is a prime Therefore, r > 17 Graduation thesis PHUONG THI ANH THU Observe that m0 m20 r= = < m0 (x2 + y2 ) ≤ · m0 m0 2 Multiplying Equation (2.3) and Equation (2.4) and applying Equation (2.1), we have m20 rp = (x21 + y12 )(x22 + y22 ) = (x1 x2 + y1 y2 )2 + (x1 y2 − x2 y1 )2 (2.5) Note that each factor in the right is divisible by m0 , since x1 x2 + y1 y2 ≡ x21 + y12 ≡ (mod m0 ), x1 y2 − x2 y1 ≡ x1 y1 − x1 y1 ≡ (mod m0 ) Now let m0 X = x1 x2 + y1 y2 and m0 Y = x1 y2 − x2 y1 Substitute these into Equation (2.5), and the dividing the obtained equation by m0 , we get rp = X + Y This means that there is r, < r < m0 , such that rp is representable as a sum of two squares This contradicts the definition of m0 Therefore, m0 = 1, and the proof is complete We next study the primality of Fermat numbers Definition 2.2.2 For each nonnegative integer n, then n-th Fermat number is defined by n Fn = 22 + 18 Graduation thesis PHUONG THI ANH THU We now consider the problem on how to know Fn is a prime or not The answer is the P´eril primality test First we recall the definition of multiplicative order Definition 2.2.3 Given an integer a and a positive integer m with (a, m) = 1, the multiplicative order of a modulo m is the smallest positive integer k with ak ≡ (mod m) The order of a modulo m is usually written as ordm (a) The following fact is well-known: Let m > and (a, m) = If ak ≡ (mod m), then ordm (a) | k In particular, ordm (a) | ϕ(n), where ϕ(n) is the Euler’s function of n Therefore, if m > and ordm (a) = m − 1, then m is a prime Theorem 2.2.4 (P´eril primality test) For n > 0, the Fermat number Fn is prime if and only if 3(Fn −1)/2 + is divisible by Fn Proof Sufficiency: assume that the congruence 3(Fn −1)/2 + ≡ (mod Fn ) holds Then, 3(Fn −1)/2 ≡ (mod Fn ) Thus, the multiplicative order n of modulo Fn divides Fn − = 22 , which is a power of On the other hand, the order does not divide (Fn − 1)/2, so the multiplicative order of modulo Fn is just Fn − Hence, Fn is a prime 19 Graduation thesis PHUONG THI ANH THU Necessity: assume that Fn is a prime By Euler’s criterion Fn 3(Fn −1)/2 ≡ (mod Fn ) n Note that 22 ≡ (mod 3) as n > By the quadratic reciprocity law we have Fn 2n = (−1)(2/2)(2 /2) · Fn = Fn = = It follows that 3(Fn −1)/2 ≡ (mod Fn ), and the proof is complete 20 Graduation thesis PHUONG THI ANH THU Conclusions In this thesis, we have presented systematically the following topics (1) Recall the definition and basic properties of congruences; (2) The definition of Quadratic residue and no-residue, and basic properties; (3) The definition of Legendre symbol and its properties; (4) State and prove the Law of Quadratic Reciprocity; (5) Some applications on represent a prime as a sum of two squares and P´eril primality test on Fermat numbers This is the first time I got acquainted with scientific research, despite many attempts, I could not avoid may shortcomings Therefore, I look forward to receiving suggestions from teachers and readers I sincerely thanks the teachers and especially Dr Tran Nam Trung has been dedicated to helping me in the past 21 Bibliography [1] I Niven, H.S Zuckerman and H L Montgomery, An Introduction to the Theory of Numbers, fifth edition, John Wiley & Sons, Inc [2] W Stein, Elementary Number Theory, A computational Approach, Springer, 2007 [3] S Wright, Quadratic residue and Non-Residue: Selected topic, Oakland University, https://arxiv.org 22 ... of two quadratic residues is a quadratic residue 2) The product of two quadratic non-residues is a quadratic residue 3) The product of a quadratic residue and a quadratic non-residue is a quadratic. .. 10 a2 b p = b p Chapter Quadratic Reciprocity The aim of of this chapter is to prove the Law of Quadratic Reciprocity and gives some its applications 2.1 Quadratic Reciprocity Lemma 2.1.1 (Gauss’s... Preface 1 Quadratic Residues 1.1 Congruences 1.2 Quadratic residues 1.3 Legendre Symbol Quadratic Reciprocity 11 2.1 Quadratic Reciprocity