Quadratic Reciprocity Angel V

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Quadratic Reciprocity Angel V A Senior Seminar On Quadratic Reciprocity Angel V. Kumchev Contents Chapter 1. Elementary Number Theory: A Review 1 1.1. Divisibility 1 1.2. The greatest common divisor of two integers 2 1.3. Prime numbers and the fundamental theorem of arithmetic 5 1.4. Congruence modulo m 5 1.5. Complete and reduced systems modulo m 8 1.6. Exercises 9 Chapter 2. The Law Of Quadratic Reciprocity 11 2.1. Polynomial congruences 11 2.2. Linear congruences 12 2.3. Quadratic congruences. Quadratic residues and nonresidues 13 2.4. The Legendre symbol 14 2.5. Quadratic reciprocity 16 2.6. Exercises 17 Chapter 3. Complex Numbers 19 3.1. Definition and algebraic properties 19 3.2. Geometric interpretation, moduli and conjugates 20 3.3. Exponential form 22 3.4. Roots of complex numbers 23 3.5. The complex exponential 23 3.6. Trigonometric and hyperbolic functions of a complex argument 24 3.7. The complex logarithm and power functions 26 3.8. Gauss sums 28 3.9. Exercises 30 Chapter 4. Algebra Over The Complex Numbers 33 4.1. Roots of polynomials with complex coefficients 33 4.2. Linear algebra over the complex numbers 35 4.3. Review of determinants 40 4.4. The resultant of two polynomials 44 4.5. Symmetric polynomials in several variables 45 4.6. Exercises 46 Chapter 5. Fourier Series 51 5.1. Definition 51 5.2. Calculus of complex-valued functions 51 5.3. An example 53 5.4. Representing functions by Fourier series 54 i ii CONTENTS 5.5. Poisson’s summation formula 56 5.6. Exercises 57 Chapter 6. Algebraic Numbers And Algebraic Integers 61 6.1. Definition 61 6.2. The ring of algebraic integers 61 6.3. Congruences between algebraic integers 63 6.4. A proof of Theorem 2.17 64 6.5. Exercises 65 Chapter 7. Proofs Of The Law Of Quadratic Reciprocity 67 7.1. S.Y. Kim’s elementary proof 67 7.2. Eisenstein’s lemma and counting lattice points 68 7.3. Another proof of Eisenstein’s 69 7.4. An algebraic proof: Gauss sums are algebraic integers 69 7.5. A proof using the formula for the Gauss sum 70 7.6. An evaluation of the Gauss sum using Fourier series 71 7.7. An evaluation of the Gauss sum using matrices 72 7.8. A proof using polynomials and resultants 73 CHAPTER 1 Elementary Number Theory: A Review As we said already in the Introduction, the focus of this course is a theorem from number theory known as the law of quadratic reciprocity. Thus, it should not come as a surprise that we shall need certain facts from elementary number theory. In this chapter, we review some of the basics of arithmetic: the definition of divisibility, the greatest common divisor of two integers, the definition and basic properties of prime numbers, and properties of congruences. It is very likely that you are familiar with much of the material in the chapter from earlier courses, so we aim mainly to refresh your knowledge, fill any potential gaps in it, and set up the notation and terminology for future reference. The pace of the exposition is thus rather brisk, and several proofs are left as exercises. 1.1. Divisibility Number theory studies the properties of the integers, and that study usually starts with the notion of divisibility of one integer by another. Definition 1.1. If a; b 2 Z and b , 0, we say that b divides a and write b j a, if a = bq for some q 2 Z. If b does not divide a, we write b - a. When b j a, we may also say that b is a divisor of a, b is a factor of a, or a is a multiple of b. Example 1.2. We have 3 j 15 and 6 j 324, but 100 - 2010. Given two integers a and b , 0, we sometimes want to divide a by b even though b may not be a divisor of a: say, we may want to divide 37 candy bars among 4 children. This problem leads to the concept of “division with a quotient and a remainder.” It is summarized by the following theorem, known as the quotient-remainder theorem or the division algorithm. Theorem 1.3 (Quotient-remainder theorem). If a; b 2 Z and b > 0, then there exist unique integers q and r such that a = bq + r; 0 ≤ r < b: Proof. “Existence.” Let q be the largest integer m with m ≤ a=b and let r = a − bq. Then q ≤ a=b =) 0 ≤ a − bq = r: Furthermore, since q is the largest integer not exceeding a=b, we have a=b < q + 1 =) a < bq + b =) r = a − bq < b: “Uniqueness.” Suppose that q1; q2; r1 and r2 are integers such that a = bq1 + r1 = bq2 + r2; 0 ≤ r1; r2 < b: Then bq1 + r1 = bq2 + r2 =) r1 − r2 = b(q2 − q1): 1 2 1. ELEMENTARY NUMBER THEORY: A REVIEW On the other hand, 0 ≤ r1; r2 < b =) −b < r1 − r2 < b: Hence, −b < b(q2 − q1) < b =) −1 < q2 − q1 < 1: Since q2 − q1 is an integer, we conclude that q2 − q1 = 0. Thus, q1 = q2 and bq1 + r1 = bq2 + r2 =) r1 = r2: 1.2. The greatest common divisor of two integers Definition 1.4. Let a and b be integers, not both 0. The greatest common divisor of a and b, denoted gcd(a; b) or (a; b), is the largest natural number d such that d j a and d j b. Example 1.5. The positive divisors of 2 are 1 and 2, and the positive divisors of 6 are 1, 2, 3 and 6. Thus, (2; 6) = 2. The positive divisors of 4 are 1, 2 and 4, and the positive divisors of 6 are 1, 2, 3 and 6. Thus, (4; 6) = 2. The positive divisors of −4 are 1, 2 and 4, and the positive divisors of 0 are all natural numbers. Thus, (−4; 0) = 4. If a , 0, then any positive divisor of a is ≤ jaj. Moreover, jaj (which is either a or −a) is a divisor of a, so jaj is the largest positive divisor of a. Thus, (a; 0) = jaj. In the above example, we computed (a; 0) for all a , 0. We want to find a method for computing the gcd of any two numbers. The next lemma reduces that problem to the case when a and b are natural numbers. Lemma 1.6. If a and b are integers and b , 0, then (a; b) = (a; jbj). Lemma 1.7. Let b; q; and r be integers, and suppose that b and r are not both 0. Then (bq + r; b) = (r; b): Proof. Let a = bq + r, d1 = (a; b), and d2 = (r; b). By the definition of (r; b), d2 j r and d2 j b, so b = d2m and r = d2n for some m; n 2 Z. Hence, a = bq + r = (d2m)q + d2n = d2(mq + n): Since mq + n is an integer, it follows that d2 j a. Since d2 is a positive divisor of b, we conclude that d2 is a natural number that divides both a and b. Then d2 does not exceed the largest natural number that divides both a and b: that is, d2 ≤ d1. Next, by the definition of (a; b), d1 j a and d1 j b, so a = d1n and b = d1m for some n; m 2 Z. Hence, r = (bq + r) − bq = a − bq = d1n − (d1m)q = d1(n − mq): Since n − mq is an integer, it follows that d1 j r. Since d1 is a positive divisor of b, we conclude that d1 is a natural number that divides both r and b. Then d1 does not exceed the largest natural number that divides both r and b: that is, d1 ≤ d2. We proved that d1 ≤ d2 and d2 ≤ d1. Therefore, d1 = d2. We can use Lemma 1.7 to calculate the gcd of any two positive integers. We illus- trate the main idea by an example, and then prove a theorem that describes the method in general. 1.2. THE GREATEST COMMON DIVISOR OF TWO INTEGERS 3 Example 1.8. Let us use Lemma 1.7 to calculate (216; 51). We apply Lemma 1.7 repeat- edly as follows: (216; 51) = (4 · 51 + 12; 51) = (12; 51) = (12; 4 · 12 + 3) = (12; 3) = 3: It seems plausible that we should be able to proceed similarly to Example 1.8 to com- pute (a; b) for general a and b. The following theorem, known as the Euclidean algorithm, establishes that this is indeed the case. Theorem 1.9 (Euclidean algorithm). Let a and b be integers, with 0 < b < a. i) If b j a, then (a; b) = b. ii) If b - a, then there exist integers q1; r1; q2; r2;:::; rm; qm+1 such that a = bq1 + r1; 0 < r1 < b; b = r1q2 + r2; 0 ≤ r2 < r1; r1 = r2q3 + r3; 0 ≤ r3 < r2; : : rm−2 = rm−1qm + rm; 0 ≤ rm < rm−1; rm−1 = rmqm+1; and (a; b) = rm. Proof. i) This part is an easy consequence of the definition of greatest common divi- sor. ii) We use induction on a to prove the statement: If b is an integer such that 0 < b < a and b - a, then there exist integers q1; r1; q2; r2;:::; rm; qm+1 such that a = bq1 + r1; 0 < r1 < b; b = r1q2 + r2; 0 ≤ r2 < r1; r1 = r2q3 + r3; 0 ≤ r3 < r2; : : rm−2 = rm−1qm + rm; 0 ≤ rm < rm−1; rm−1 = rmqm+1; rm = (a; b): The base case of the induction is a = 3.
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