Introduction to Arithmetic Geometry Fall 2013 Lecture #10 10/8/2013

Size: px
Start display at page:

Download "Introduction to Arithmetic Geometry Fall 2013 Lecture #10 10/8/2013"

Transcription

1 Introduction to Arithmetic Geometry Fall 2013 Lecture #10 10/8/2013 In this lecture we lay the groundwork needed to rove the Hasse-Minkowski theorem for Q, which states that a quadratic form over Q reresents 0 if and only if it reresents 0 over every comletion of Q (as roved by Minkowski). The statement still holds if Q is relaced by any number field (as roved by Hasse), but we will restrict our attention to Q. Unless otherwise indicated, we use througout to denote any rime of Q, including the archimedean rime =. We begin by defining the Hilbert symbol for The Hilbert symbol Definition For a, b Q the Hilbert symbol (a, b) is defined by { 1 ax 2 + by 2 = 1 has a solution in Q, (a, b) = 1 otherwise. It is clear from the definition that the Hilbert symbol is symmetric, and that it only deends on the images of a and b in Q /Q 2 (their square classes). We note that Q /Q 2 Z/2Z if =, (Z/2Z )2 if is odd, (Z/2Z) 3 if = 2. The case = is clear, since R = Q has just two square classes (ositive and negative numbers), and the cases with < were roved in Problem Set 4. Thus the Hilbert symbol can be viewed as a ma (Q /Q 2 ) (Q /Q 2 ) {±1} of finite sets. We say that a solution (x 0,..., x n ) to a homogeneous olynomial equation over Q is rimitive if all of its elements lie in Z and at least one lies in Z. The following lemma gives several equivalent definitions of the Hilbert symbol. Lemma For any a, b Q, the following are equivalent: (i) (a, b) = 1. (ii) The quadratic form z 2 ax 2 by 2 reresents 0. (iii) The equation ax 2 + by 2 = z 2 has a rimitive solution. (iv) a Q is the norm of an element in Q ( b). Proof. (i) (ii) is immediate (let z = 1). The reverse imlication is clear if z 2 ax 2 by 2 = 0 reresents 0 with z nonzero (divide by z 2 ), and otherwise the non-degenerate quadratic form ax 2 + by 2 reresents 0, hence it reresents every element of Q including 1, so (ii) (i). To show (ii) (iii), multily through by r, for a suitable integer r, and rearrange terms. The reverse imlication (iii) (ii) is immediate. If b is square then Q ( b) = Q and N(a) = a so (iv) holds, and the form z 2 by 2 reresents 0, hence every element of Q including ax 2 0 for any x 0, so (ii) holds. If b is not square then N(z +y b) = z 2 by ( 2. If a is a norm in Q b) then z 2 ax 2 by 2 reresents 0 (set x = 1), and if z 2 ax 2 by 2 reresents 0 then dividing by x 2 and adding a to both sides shows that a is a norm. So (ii) (iv). 1 Andrew V. Sutherland

2 Corollary For all a, b, c Q, the following hold: (i) (1, c) = 1. (ii) ( c, c) = 1. (iii) (a, c) = 1 = (a, c) (b, c) = (ab, c). (iv) (c, c) = ( 1, c). Proof. Let N denote the norm ma from Q ( c) to Q. For (i) we have N(1) = 1. For (ii), c = N( c) for c Q 2 and c = N( c) otherwise. For (iii), If a and b are both norms in Q( c), then so is ab, by the multilicativity of the norm ma; conversely, if a and ab are both norms, so is 1/a, as is (1/a)ab = b. Thus if (a, c) = 1, then (b, c) = 1 if and only if (ab, c) = 1, which imlies (a, c) (b, c) = (ab, c). For (iv), ( c, c) = 1 by (ii), so by (iii) we have (c, c) = ( c, c) (c, c) = ( c 2, c) = ( 1, c). Theorem (a, b) = 1 if and only if a, b < 0 Proof. We can assume a, b {±1}, since {±1} is a comlete set of reresentatives for R /R 2. If either a or b is 1 then ( a, b) = 1, by Corollary 10.3.(i), and ( 1, 1) = 1, since 1 is not a norm in C = Q ( 1). Lemma If is odd, then (u, v) = 1 for all u, v Z. Proof. Recall from Lecture 3 (or the Chevalley-Warning theorem on roblem set 2) that every lane rojective conic over F has a rational oint, so we can find a non-trivial solution to z 2 ux 2 vy 2 = 0 modulo. If we then fix two of x, y, z so that the third is nonzero, Hensel s lemma gives a solution over Z. Remark Lemma 10.5 does not hold for = 2; for examle, (3, 3) 2 = 1. Theorem Let be an odd rime, and write a, b Q α, β Z and u, v Z. Then as a = α u and b = β v, with (a, b) = ( 1) ( β 1 αβ 2 x mod where ( x ) denotes the Legendre symbol ( ). u ) ( v Proof. Since (a, b) deends only on the square classes of a and b, we assume α, β {0, 1}. Case α = 0, β = 0: We have (u, v) = 1, by Lemma 10.5, which agrees with the formula. Case α = 1, β = 0: We need to show that (u, v) = ( v ). Since (u 1, v) = 1, we have (u, v) = (u, v) (u 1, v) = (, v), by Corollary 10.3.(iii). If v is a square then we have (, v) = (, 1) = (1, ) = 1 = ( v ). If v is not a square then z 2 x 2 vy 2 = 0 has no non- v trivial solutions modulo, hence no rimitive solutions. This imlies ( ) ( (, v) = 1 = ( ) ). 1 Case α = 1, β = 1: We must show (u, v) u v = ( 1) 2. Alying Corollary 10.3 we have ) α, (u, v) = (u, v) ( v, v) = ( 2 uv, v) = ( uv, v) = (v, uv) Alying the formula in the case α = 1, β = 0 already roved, we have ( ) ( ) ( ) ( ) ( uv 1 u v u (v, uv) = = = ( 1) 1 2 ) ( v ). 2

3 Lemma Let u, v Z. The equations z 2 ux 2 vy 2 = 0 and z 2 2ux vy = 0 have rimitive solutions over Z 2 if and only if they have rimitive solutions modulo 8. Proof. Without loss of generality we can assume that u and v are odd integers, since every square class in Z 2 /Z 2 2 is reresented by an odd integer (in fact one can assume u, v {±1, ±5}) The necessity of having a rimitive solution modulo 8 is clear. To rove sufficiency we aly the strong form of Hensel s lemma roved in Problem Set 4. In both cases, if we have a non-trivial solution (x 0, y 0, z 0 ) modulo 8 we can fix two of x 0, y 0, z 0 to obtain a quadratic olynomial f(w) over Z 2 and w 0 Z 2 that satisfies v 2(f(w 0 )) = 3 > 2 = 2v 2 (f (w 0 )). In the case of the second equation, note that a rimitive solution (x 0, y 0, z 0 ) modulo 8 must have y 0 or z 0 odd; if not, then z0 2 and vy2 0, and therefore 2ux2 0, are divisible by 4, but this means x 0 is also divisible by 2, which contradicts the rimitivity of (x 0, y 0, z 0 ). Lifting w 0 to a root of f(w) over Z 2 yields a solution to the original equation. Theorem Write a, b Q 2 as a = 2α u and b = 2 β v with α, β Z and u, v Z 2. Then (a, b) 2 = ( 1) ɛ(u)ɛ(v)+αω(v)+βω(u), where ɛ(u) and ω(u) denote the images in Z/2Z of (u 1)/2 and (u 2 1)/8, resectively. Proof. Since (a, b) 2 only deends on the square classes of a and b, It suffices to verify the formula for a, b S, where S = {±1, ±3, ±2, ±6} is a comlete set of reresentatives for Q 2 /Q 2 2. As in the roof of Theorem 10.7, we can use (u, v) 2 = (v, uv) 2 to reduce to the case where one of a, b lies in Z. By Lemma 10.8, to comute (a, b) 2 with one of a, b in Z 2, it suffices to check for rimitive solutions to z 2 ax 2 by 2 = 0 modulo 8, which reduces the roblem to a finite verification which erformed by We now note the following corollary to Theorems 10.4, 10.7, and Corollary The Hilbert symbol (a, b) is a nondegenerate bilinear ma. This means that for all a, b, c Q we have (a, c) (b, c) = (ab, c) and (a, b) (a, c) = (a, bc), and that for every non-square c we have (b, c) = 1 for some b. Proof. Both statements are clear for = (there are only 2 square classes and 4 combi 1 nations to check). For odd, let c = γ w and fix ε = ( 1) γ 2. Then for a = α u and b = β v, we have ( ) ( ) ( ) ( ) α u w β v w ( a, c) (b, c) = ε ε ( ) γ ( ) uv w α+β = ε α+β = (ab, c). γ α γ β To verify non-degeneracy, we note that if c is not square than either γ = 1 or ( w ) = 1. If γ γ = 1 we can choose b = v with ( v ) = 1, so that (b, c) = ( v ) = 1. If γ = 0, then ε = 1 and ( w ) = 1, so withb = we have (b, c) = ( w ) = 1. 3

4 For = 2, we have (a, c) (b, c) = ( 1) ɛ(u)ɛ(w)+αω(w)+γω(u) ( 1) ɛ(v)ɛ(w)+βω(w)+γω(v) 2 2 = ( 1) (ɛ(u)+ɛ(v))ɛ(w)+(α+β)ω(w)+γ(ω(u)+ω(v)) = ( 1) ɛ(uv)ɛ(w)+(α+β)ω(w)+γω(uv) = (ab, c) 2, where we have used the fact that ɛ and ω are grou homomorhisms from Z 2 to Z/2Z. To see this, note that the image of ɛ 1 (0) in (Z/4Z) is {1}, a subgrou of index 2, and the image of ω 1 (0) in (Z/8Z) is {±1}, which is again a subgrou of index 2. We now verify non-degeneracy for = 2. If c is not square then either γ = 1, or one of ɛ(w) and ω(w) is nonzero. If γ = 1, then (5, c) 2 = 1. If γ = 0 and ω(w) = 1, then (2, c) 2 = 1. If γ = 0 and ω(w) = 0, then we must have ɛ(w) = 1, so ( 1, c) 2 = 1. We now rove Hilbert s recirocity law, which may be regarded as a generalization of quadratic recirocity. Theorem Let a, b Q. Then (a, b) = 1 for all but finitely many rimes and (a, b) = 1. Proof. We can assume without loss of generality that a, b Z, since multilying each of a and b by the square of its denominator will not change (a, b) for any. The theorem holds if either a or b is 1, and by the bilinearity of the Hilbert symbol, we can assume that a, b { 1} {q Z >0 : q is rime}. The first statement of the theorem is clear, since a, b Z for < not equal to a or b, and (u, v) = 1 for all u, v Z when is odd, by Lemma To verify the roduct formula, we consider 5 cases. Case 1: a = b = 1. Then ( 1, 1) = ( 1, 1) 2 = 1 and ( 1, 1) = 1 for odd. Case 2: a = 1 and b is rime. If b = 2 then (1, 1) is a solution to x 2 + 2y 2 = 1 over Q for all, thus ( 1, 2) = 1. If b is odd, then ( 1, b) = 1 for {2, b}, while ɛ(b) 1 ( 1, b) (b 1) 2 = ( 1) and ( 1, b) b = ( b ), both of which are equal to ( 1) /2. Case 3: a and b are the same rime. Then by Corollary 10.3, (b, b) = ( 1, b) for all rimes, and we are in case 2. Case 4: a = 2 and b is an odd rime. Then (2, b) = 1 for all {2, b}, while ω(b) 2 (2, b) (b 2 1)/8 2 = ( 1) and (2, b) b = ( ), both of which are equal to ( 1). Case 5: a and b are distinct odd rimes. Then (a, b) = 1 for all {2, a, b}, while ( 1) ɛ(a)ɛ(b) if = 2, ( ) (a, b) a = ( b if = b, b ) if = a. Since ɛ(x) = (x 1)/2 mod 2, we have ( a 1 b 1 a ) ( b ) (a, b) = ( 1) 2 2 = 1, b a by quadratic recirocity. a 4

5 MIT OenCourseWare htt://ocw.mit.edu 201 For information about citing these materials or our Terms of Use, visit: htt://ocw.mit.edu/terms.

The Hasse Minkowski Theorem Lee Dicker University of Minnesota, REU Summer 2001

The Hasse Minkowski Theorem Lee Dicker University of Minnesota, REU Summer 2001 The Hasse Minkowski Theorem Lee Dicker University of Minnesota, REU Summer 2001 The Hasse-Minkowski Theorem rovides a characterization of the rational quadratic forms. What follows is a roof of the Hasse-Minkowski

More information

Quadratic Residues, Quadratic Reciprocity. 2 4 So we may as well start with x 2 a mod p. p 1 1 mod p a 2 ±1 mod p

Quadratic Residues, Quadratic Reciprocity. 2 4 So we may as well start with x 2 a mod p. p 1 1 mod p a 2 ±1 mod p Lecture 9 Quadratic Residues, Quadratic Recirocity Quadratic Congruence - Consider congruence ax + bx + c 0 mod, with a 0 mod. This can be reduced to x + ax + b 0, if we assume that is odd ( is trivial

More information

3 Properties of Dedekind domains

3 Properties of Dedekind domains 18.785 Number theory I Fall 2016 Lecture #3 09/15/2016 3 Proerties of Dedekind domains In the revious lecture we defined a Dedekind domain as a noetherian domain A that satisfies either of the following

More information

Jacobi symbols and application to primality

Jacobi symbols and application to primality Jacobi symbols and alication to rimality Setember 19, 018 1 The grou Z/Z We review the structure of the abelian grou Z/Z. Using Chinese remainder theorem, we can restrict to the case when = k is a rime

More information

Math 4400/6400 Homework #8 solutions. 1. Let P be an odd integer (not necessarily prime). Show that modulo 2,

Math 4400/6400 Homework #8 solutions. 1. Let P be an odd integer (not necessarily prime). Show that modulo 2, MATH 4400 roblems. Math 4400/6400 Homework # solutions 1. Let P be an odd integer not necessarily rime. Show that modulo, { P 1 0 if P 1, 7 mod, 1 if P 3, mod. Proof. Suose that P 1 mod. Then we can write

More information

x 2 a mod m. has a solution. Theorem 13.2 (Euler s Criterion). Let p be an odd prime. The congruence x 2 1 mod p,

x 2 a mod m. has a solution. Theorem 13.2 (Euler s Criterion). Let p be an odd prime. The congruence x 2 1 mod p, 13. Quadratic Residues We now turn to the question of when a quadratic equation has a solution modulo m. The general quadratic equation looks like ax + bx + c 0 mod m. Assuming that m is odd or that b

More information

RECIPROCITY LAWS JEREMY BOOHER

RECIPROCITY LAWS JEREMY BOOHER RECIPROCITY LAWS JEREMY BOOHER 1 Introduction The law of uadratic recirocity gives a beautiful descrition of which rimes are suares modulo Secial cases of this law going back to Fermat, and Euler and Legendre

More information

MA3H1 TOPICS IN NUMBER THEORY PART III

MA3H1 TOPICS IN NUMBER THEORY PART III MA3H1 TOPICS IN NUMBER THEORY PART III SAMIR SIKSEK 1. Congruences Modulo m In quadratic recirocity we studied congruences of the form x 2 a (mod ). We now turn our attention to situations where is relaced

More information

QUADRATIC RECIPROCITY

QUADRATIC RECIPROCITY QUADRATIC RECIPROCITY JORDAN SCHETTLER Abstract. The goals of this roject are to have the reader(s) gain an areciation for the usefulness of Legendre symbols and ultimately recreate Eisenstein s slick

More information

MATH342 Practice Exam

MATH342 Practice Exam MATH342 Practice Exam This exam is intended to be in a similar style to the examination in May/June 2012. It is not imlied that all questions on the real examination will follow the content of the ractice

More information

We collect some results that might be covered in a first course in algebraic number theory.

We collect some results that might be covered in a first course in algebraic number theory. 1 Aendices We collect some results that might be covered in a first course in algebraic number theory. A. uadratic Recirocity Via Gauss Sums A1. Introduction In this aendix, is an odd rime unless otherwise

More information

QUADRATIC RECIPROCITY

QUADRATIC RECIPROCITY QUADRATIC RECIPROCITY JORDAN SCHETTLER Abstract. The goals of this roject are to have the reader(s) gain an areciation for the usefulness of Legendre symbols and ultimately recreate Eisenstein s slick

More information

QUADRATIC FORMS, BASED ON (A COURSE IN ARITHMETIC BY SERRE)

QUADRATIC FORMS, BASED ON (A COURSE IN ARITHMETIC BY SERRE) QUADRATIC FORMS, BASED ON (A COURSE IN ARITHMETIC BY SERRE) HEE OH 1. Lecture 1:Introduction and Finite fields Let f be a olynomial with integer coefficients. One of the basic roblem is to understand if

More information

MATH 361: NUMBER THEORY EIGHTH LECTURE

MATH 361: NUMBER THEORY EIGHTH LECTURE MATH 361: NUMBER THEORY EIGHTH LECTURE 1. Quadratic Recirocity: Introduction Quadratic recirocity is the first result of modern number theory. Lagrange conjectured it in the late 1700 s, but it was first

More information

MATH 361: NUMBER THEORY ELEVENTH LECTURE

MATH 361: NUMBER THEORY ELEVENTH LECTURE MATH 361: NUMBER THEORY ELEVENTH LECTURE The subjects of this lecture are characters, Gauss sums, Jacobi sums, and counting formulas for olynomial equations over finite fields. 1. Definitions, Basic Proerties

More information

Mobius Functions, Legendre Symbols, and Discriminants

Mobius Functions, Legendre Symbols, and Discriminants Mobius Functions, Legendre Symbols, and Discriminants 1 Introduction Zev Chonoles, Erick Knight, Tim Kunisky Over the integers, there are two key number-theoretic functions that take on values of 1, 1,

More information

POINTS ON CONICS MODULO p

POINTS ON CONICS MODULO p POINTS ON CONICS MODULO TEAM 2: JONGMIN BAEK, ANAND DEOPURKAR, AND KATHERINE REDFIELD Abstract. We comute the number of integer oints on conics modulo, where is an odd rime. We extend our results to conics

More information

Weil s Conjecture on Tamagawa Numbers (Lecture 1)

Weil s Conjecture on Tamagawa Numbers (Lecture 1) Weil s Conjecture on Tamagawa Numbers (Lecture ) January 30, 204 Let R be a commutative ring and let V be an R-module. A quadratic form on V is a ma q : V R satisfying the following conditions: (a) The

More information

HENSEL S LEMMA KEITH CONRAD

HENSEL S LEMMA KEITH CONRAD HENSEL S LEMMA KEITH CONRAD 1. Introduction In the -adic integers, congruences are aroximations: for a and b in Z, a b mod n is the same as a b 1/ n. Turning information modulo one ower of into similar

More information

Quadratic Reciprocity

Quadratic Reciprocity Quadratic Recirocity 5-7-011 Quadratic recirocity relates solutions to x = (mod to solutions to x = (mod, where and are distinct odd rimes. The euations are oth solvale or oth unsolvale if either or has

More information

6 Binary Quadratic forms

6 Binary Quadratic forms 6 Binary Quadratic forms 6.1 Fermat-Euler Theorem A binary quadratic form is an exression of the form f(x,y) = ax 2 +bxy +cy 2 where a,b,c Z. Reresentation of an integer by a binary quadratic form has

More information

MATH 3240Q Introduction to Number Theory Homework 7

MATH 3240Q Introduction to Number Theory Homework 7 As long as algebra and geometry have been searated, their rogress have been slow and their uses limited; but when these two sciences have been united, they have lent each mutual forces, and have marched

More information

Math 261 Exam 2. November 7, The use of notes and books is NOT allowed.

Math 261 Exam 2. November 7, The use of notes and books is NOT allowed. Math 261 Eam 2 ovember 7, 2018 The use of notes and books is OT allowed Eercise 1: Polynomials mod 691 (30 ts In this eercise, you may freely use the fact that 691 is rime Consider the olynomials f( 4

More information

Solvability and Number of Roots of Bi-Quadratic Equations over p adic Fields

Solvability and Number of Roots of Bi-Quadratic Equations over p adic Fields Malaysian Journal of Mathematical Sciences 10(S February: 15-35 (016 Secial Issue: The 3 rd International Conference on Mathematical Alications in Engineering 014 (ICMAE 14 MALAYSIAN JOURNAL OF MATHEMATICAL

More information

Algebraic number theory LTCC Solutions to Problem Sheet 2

Algebraic number theory LTCC Solutions to Problem Sheet 2 Algebraic number theory LTCC 008 Solutions to Problem Sheet ) Let m be a square-free integer and K = Q m). The embeddings K C are given by σ a + b m) = a + b m and σ a + b m) = a b m. If m mod 4) then

More information

Elementary Analysis in Q p

Elementary Analysis in Q p Elementary Analysis in Q Hannah Hutter, May Szedlák, Phili Wirth November 17, 2011 This reort follows very closely the book of Svetlana Katok 1. 1 Sequences and Series In this section we will see some

More information

A CONCRETE EXAMPLE OF PRIME BEHAVIOR IN QUADRATIC FIELDS. 1. Abstract

A CONCRETE EXAMPLE OF PRIME BEHAVIOR IN QUADRATIC FIELDS. 1. Abstract A CONCRETE EXAMPLE OF PRIME BEHAVIOR IN QUADRATIC FIELDS CASEY BRUCK 1. Abstract The goal of this aer is to rovide a concise way for undergraduate mathematics students to learn about how rime numbers behave

More information

DISCRIMINANTS IN TOWERS

DISCRIMINANTS IN TOWERS DISCRIMINANTS IN TOWERS JOSEPH RABINOFF Let A be a Dedekind domain with fraction field F, let K/F be a finite searable extension field, and let B be the integral closure of A in K. In this note, we will

More information

t s (p). An Introduction

t s (p). An Introduction Notes 6. Quadratic Gauss Sums Definition. Let a, b Z. Then we denote a b if a divides b. Definition. Let a and b be elements of Z. Then c Z s.t. a, b c, where c gcda, b max{x Z x a and x b }. 5, Chater1

More information

arxiv: v2 [math.nt] 11 Jun 2016

arxiv: v2 [math.nt] 11 Jun 2016 Congruent Ellitic Curves with Non-trivial Shafarevich-Tate Grous Zhangjie Wang Setember 18, 018 arxiv:1511.03810v [math.nt 11 Jun 016 Abstract We study a subclass of congruent ellitic curves E n : y x

More information

HOMEWORK # 4 MARIA SIMBIRSKY SANDY ROGERS MATTHEW WELSH

HOMEWORK # 4 MARIA SIMBIRSKY SANDY ROGERS MATTHEW WELSH HOMEWORK # 4 MARIA SIMBIRSKY SANDY ROGERS MATTHEW WELSH 1. Section 2.1, Problems 5, 8, 28, and 48 Problem. 2.1.5 Write a single congruence that is equivalent to the air of congruences x 1 mod 4 and x 2

More information

MAT 311 Solutions to Final Exam Practice

MAT 311 Solutions to Final Exam Practice MAT 311 Solutions to Final Exam Practice Remark. If you are comfortable with all of the following roblems, you will be very well reared for the midterm. Some of the roblems below are more difficult than

More information

PROBLEM SET 5 SOLUTIONS. Solution. We prove that the given congruence equation has no solutions. Suppose for contradiction that. (x 2) 2 1 (mod 7).

PROBLEM SET 5 SOLUTIONS. Solution. We prove that the given congruence equation has no solutions. Suppose for contradiction that. (x 2) 2 1 (mod 7). PROBLEM SET 5 SOLUTIONS 1 Fid every iteger solutio to x 17x 5 0 mod 45 Solutio We rove that the give cogruece equatio has o solutios Suose for cotradictio that the equatio x 17x 5 0 mod 45 has a solutio

More information

By Evan Chen OTIS, Internal Use

By Evan Chen OTIS, Internal Use Solutions Notes for DNY-NTCONSTRUCT Evan Chen January 17, 018 1 Solution Notes to TSTST 015/5 Let ϕ(n) denote the number of ositive integers less than n that are relatively rime to n. Prove that there

More information

Practice Final Solutions

Practice Final Solutions Practice Final Solutions 1. Find integers x and y such that 13x + 1y 1 SOLUTION: By the Euclidean algorithm: One can work backwards to obtain 1 1 13 + 2 13 6 2 + 1 1 13 6 2 13 6 (1 1 13) 7 13 6 1 Hence

More information

Practice Final Solutions

Practice Final Solutions Practice Final Solutions 1. True or false: (a) If a is a sum of three squares, and b is a sum of three squares, then so is ab. False: Consider a 14, b 2. (b) No number of the form 4 m (8n + 7) can be written

More information

f(r) = a d n) d + + a0 = 0

f(r) = a d n) d + + a0 = 0 Math 400-00/Foundations of Algebra/Fall 07 Polynomials at the Foundations: Roots Next, we turn to the notion of a root of a olynomial in Q[x]. Definition 8.. r Q is a rational root of fx) Q[x] if fr) 0.

More information

(Workshop on Harmonic Analysis on symmetric spaces I.S.I. Bangalore : 9th July 2004) B.Sury

(Workshop on Harmonic Analysis on symmetric spaces I.S.I. Bangalore : 9th July 2004) B.Sury Is e π 163 odd or even? (Worksho on Harmonic Analysis on symmetric saces I.S.I. Bangalore : 9th July 004) B.Sury e π 163 = 653741640768743.999999999999.... The object of this talk is to exlain this amazing

More information

Multiplicative group law on the folium of Descartes

Multiplicative group law on the folium of Descartes Multilicative grou law on the folium of Descartes Steluţa Pricoie and Constantin Udrişte Abstract. The folium of Descartes is still studied and understood today. Not only did it rovide for the roof of

More information

#A64 INTEGERS 18 (2018) APPLYING MODULAR ARITHMETIC TO DIOPHANTINE EQUATIONS

#A64 INTEGERS 18 (2018) APPLYING MODULAR ARITHMETIC TO DIOPHANTINE EQUATIONS #A64 INTEGERS 18 (2018) APPLYING MODULAR ARITHMETIC TO DIOPHANTINE EQUATIONS Ramy F. Taki ElDin Physics and Engineering Mathematics Deartment, Faculty of Engineering, Ain Shams University, Cairo, Egyt

More information

LECTURE 10: JACOBI SYMBOL

LECTURE 10: JACOBI SYMBOL LECTURE 0: JACOBI SYMBOL The Jcobi symbol We wish to generlise the Legendre symbol to ccomodte comosite moduli Definition Let be n odd ositive integer, nd suose tht s, where the i re rime numbers not necessrily

More information

Math 104B: Number Theory II (Winter 2012)

Math 104B: Number Theory II (Winter 2012) Math 104B: Number Theory II (Winter 01) Alina Bucur Contents 1 Review 11 Prime numbers 1 Euclidean algorithm 13 Multilicative functions 14 Linear diohantine equations 3 15 Congruences 3 Primes as sums

More information

MAZUR S CONSTRUCTION OF THE KUBOTA LEPOLDT p-adic L-FUNCTION. n s = p. (1 p s ) 1.

MAZUR S CONSTRUCTION OF THE KUBOTA LEPOLDT p-adic L-FUNCTION. n s = p. (1 p s ) 1. MAZUR S CONSTRUCTION OF THE KUBOTA LEPOLDT -ADIC L-FUNCTION XEVI GUITART Abstract. We give an overview of Mazur s construction of the Kubota Leooldt -adic L- function as the -adic Mellin transform of a

More information

CONGRUENCES CONCERNING LUCAS SEQUENCES ZHI-HONG SUN

CONGRUENCES CONCERNING LUCAS SEQUENCES ZHI-HONG SUN Int. J. Number Theory 004, no., 79-85. CONGRUENCES CONCERNING LUCAS SEQUENCES ZHI-HONG SUN School of Mathematical Sciences Huaiyin Normal University Huaian, Jiangsu 00, P.R. China zhihongsun@yahoo.com

More information

On Character Sums of Binary Quadratic Forms 1 2. Mei-Chu Chang 3. Abstract. We establish character sum bounds of the form.

On Character Sums of Binary Quadratic Forms 1 2. Mei-Chu Chang 3. Abstract. We establish character sum bounds of the form. On Character Sums of Binary Quadratic Forms 2 Mei-Chu Chang 3 Abstract. We establish character sum bounds of the form χ(x 2 + ky 2 ) < τ H 2, a x a+h b y b+h where χ is a nontrivial character (mod ), 4

More information

Representing Integers as the Sum of Two Squares in the Ring Z n

Representing Integers as the Sum of Two Squares in the Ring Z n 1 2 3 47 6 23 11 Journal of Integer Sequences, Vol. 17 (2014), Article 14.7.4 Reresenting Integers as the Sum of Two Squares in the Ring Z n Joshua Harrington, Lenny Jones, and Alicia Lamarche Deartment

More information

Legendre polynomials and Jacobsthal sums

Legendre polynomials and Jacobsthal sums Legendre olynomials and Jacobsthal sums Zhi-Hong Sun( Huaiyin Normal University( htt://www.hytc.edu.cn/xsjl/szh Notation: Z the set of integers, N the set of ositive integers, [x] the greatest integer

More information

CERIAS Tech Report The period of the Bell numbers modulo a prime by Peter Montgomery, Sangil Nahm, Samuel Wagstaff Jr Center for Education

CERIAS Tech Report The period of the Bell numbers modulo a prime by Peter Montgomery, Sangil Nahm, Samuel Wagstaff Jr Center for Education CERIAS Tech Reort 2010-01 The eriod of the Bell numbers modulo a rime by Peter Montgomery, Sangil Nahm, Samuel Wagstaff Jr Center for Education and Research Information Assurance and Security Purdue University,

More information

Chapter 2 Arithmetic Functions and Dirichlet Series.

Chapter 2 Arithmetic Functions and Dirichlet Series. Chater 2 Arithmetic Functions and Dirichlet Series. [4 lectures] Definition 2.1 An arithmetic function is any function f : N C. Examles 1) The divisor function d (n) (often denoted τ (n)) is the number

More information

On the Rank of the Elliptic Curve y 2 = x(x p)(x 2)

On the Rank of the Elliptic Curve y 2 = x(x p)(x 2) On the Rank of the Ellitic Curve y = x(x )(x ) Jeffrey Hatley Aril 9, 009 Abstract An ellitic curve E defined over Q is an algebraic variety which forms a finitely generated abelian grou, and the structure

More information

QUADRATIC RECIPROCITY

QUADRATIC RECIPROCITY QUADRATIC RECIPROCIT POOJA PATEL Abstract. This aer is an self-contained exosition of the law of uadratic recirocity. We will give two roofs of the Chinese remainder theorem and a roof of uadratic recirocity.

More information

Factor Rings and their decompositions in the Eisenstein integers Ring Z [ω]

Factor Rings and their decompositions in the Eisenstein integers Ring Z [ω] Armenian Journal of Mathematics Volume 5, Number 1, 013, 58 68 Factor Rings and their decomositions in the Eisenstein integers Ring Z [ω] Manouchehr Misaghian Deartment of Mathematics, Prairie View A&M

More information

RINGS OF INTEGERS WITHOUT A POWER BASIS

RINGS OF INTEGERS WITHOUT A POWER BASIS RINGS OF INTEGERS WITHOUT A POWER BASIS KEITH CONRAD Let K be a number field, with degree n and ring of integers O K. When O K = Z[α] for some α O K, the set {1, α,..., α n 1 } is a Z-basis of O K. We

More information

Algebraic Number Theory

Algebraic Number Theory Algebraic Number Theory Joseh R. Mileti May 11, 2012 2 Contents 1 Introduction 5 1.1 Sums of Squares........................................... 5 1.2 Pythagorean Triles.........................................

More information

Math 751 Lecture Notes Week 3

Math 751 Lecture Notes Week 3 Math 751 Lecture Notes Week 3 Setember 25, 2014 1 Fundamental grou of a circle Theorem 1. Let φ : Z π 1 (S 1 ) be given by n [ω n ], where ω n : I S 1 R 2 is the loo ω n (s) = (cos(2πns), sin(2πns)). Then

More information

QUADRATIC RESIDUES AND DIFFERENCE SETS

QUADRATIC RESIDUES AND DIFFERENCE SETS QUADRATIC RESIDUES AND DIFFERENCE SETS VSEVOLOD F. LEV AND JACK SONN Abstract. It has been conjectured by Sárközy that with finitely many excetions, the set of quadratic residues modulo a rime cannot be

More information

When do Fibonacci invertible classes modulo M form a subgroup?

When do Fibonacci invertible classes modulo M form a subgroup? Calhoun: The NPS Institutional Archive DSace Reository Faculty and Researchers Faculty and Researchers Collection 2013 When do Fibonacci invertible classes modulo M form a subgrou? Luca, Florian Annales

More information

MA257: INTRODUCTION TO NUMBER THEORY LECTURE NOTES 2018

MA257: INTRODUCTION TO NUMBER THEORY LECTURE NOTES 2018 MA257: INTRODUCTION TO NUMBER THEORY LECTURE NOTES 2018 J. E. CREMONA Contents 0. Introduction: What is Number Theory? 2 Basic Notation 3 1. Factorization 4 1.1. Divisibility in Z 4 1.2. Greatest Common

More information

1 Integers and the Euclidean algorithm

1 Integers and the Euclidean algorithm 1 1 Integers and the Euclidean algorithm Exercise 1.1 Prove, n N : induction on n) 1 3 + 2 3 + + n 3 = (1 + 2 + + n) 2 (use Exercise 1.2 Prove, 2 n 1 is rime n is rime. (The converse is not true, as shown

More information

Primes - Problem Sheet 5 - Solutions

Primes - Problem Sheet 5 - Solutions Primes - Problem Sheet 5 - Solutions Class number, and reduction of quadratic forms Positive-definite Q1) Aly the roof of Theorem 5.5 to find reduced forms equivalent to the following, also give matrices

More information

HASSE INVARIANTS FOR THE CLAUSEN ELLIPTIC CURVES

HASSE INVARIANTS FOR THE CLAUSEN ELLIPTIC CURVES HASSE INVARIANTS FOR THE CLAUSEN ELLIPTIC CURVES AHMAD EL-GUINDY AND KEN ONO Astract. Gauss s F x hyergeometric function gives eriods of ellitic curves in Legendre normal form. Certain truncations of this

More information

RECIPROCITY, BRAUER GROUPS AND QUADRATIC FORMS OVER NUMBER FIELDS

RECIPROCITY, BRAUER GROUPS AND QUADRATIC FORMS OVER NUMBER FIELDS RECIPROCITY, BRAUER GROUPS AND QUADRATIC FORMS OVER NUMBER FIELDS THOMAS POGUNTKE Abstract. Artin s recirocity law is a vast generalization of quadratic recirocity and contains a lot of information about

More information

ORBIT OF QUADRATIC IRRATIONALS MODULO P BY THE MODULAR GROUP ABSTRACT 2

ORBIT OF QUADRATIC IRRATIONALS MODULO P BY THE MODULAR GROUP ABSTRACT 2 ORBIT OF QUADRATIC IRRATIONALS MODULO P B THE MODULAR GROUP Shin-Ichi Katayama, Toru Nakahara, Syed Inayat Ali Shah 3, Mohammad Naeem Khalid 3 and Sareer Badshah 3 Tokushima University, Jaan. Saga University,

More information

Analytic number theory and quadratic reciprocity

Analytic number theory and quadratic reciprocity Analytic number theory and quadratic recirocity Levent Aloge March 31, 013 Abstract What could the myriad tools of analytic number theory for roving bounds on oscillating sums ossibly have to say about

More information

MATH 371 Class notes/outline September 24, 2013

MATH 371 Class notes/outline September 24, 2013 MATH 371 Class notes/outline Setember 24, 2013 Rings Armed with what we have looked at for the integers and olynomials over a field, we re in a good osition to take u the general theory of rings. Definitions:

More information

Elliptic Curves Spring 2015 Problem Set #1 Due: 02/13/2015

Elliptic Curves Spring 2015 Problem Set #1 Due: 02/13/2015 18.783 Ellitic Curves Sring 2015 Problem Set #1 Due: 02/13/2015 Descrition These roblems are related to the material covered in Lectures 1-2. Some of them require the use of Sage, and you will need to

More information

Class Field Theory. Peter Stevenhagen. 1. Class Field Theory for Q

Class Field Theory. Peter Stevenhagen. 1. Class Field Theory for Q Class Field Theory Peter Stevenhagen Class field theory is the study of extensions Q K L K ab K = Q, where L/K is a finite abelian extension with Galois grou G. 1. Class Field Theory for Q First we discuss

More information

DIRICHLET S THEOREM ABOUT PRIMES IN ARITHMETIC PROGRESSIONS. Contents. 1. Dirichlet s theorem on arithmetic progressions

DIRICHLET S THEOREM ABOUT PRIMES IN ARITHMETIC PROGRESSIONS. Contents. 1. Dirichlet s theorem on arithmetic progressions DIRICHLET S THEOREM ABOUT PRIMES IN ARITHMETIC PROGRESSIONS ANG LI Abstract. Dirichlet s theorem states that if q and l are two relatively rime ositive integers, there are infinitely many rimes of the

More information

Almost All Palindromes Are Composite

Almost All Palindromes Are Composite Almost All Palindromes Are Comosite William D Banks Det of Mathematics, University of Missouri Columbia, MO 65211, USA bbanks@mathmissouriedu Derrick N Hart Det of Mathematics, University of Missouri Columbia,

More information

LECTURE 2. Hilbert Symbols

LECTURE 2. Hilbert Symbols LECTURE 2 Hilbert Symbols Let be a local field over Q p (though any local field suffices) with char() 2. Note that this includes fields over Q 2, since it is the characteristic of the field, and not the

More information

16 The Quadratic Reciprocity Law

16 The Quadratic Reciprocity Law 16 The Quadratic Recirocity Law Fix an odd rime If is another odd rime, a fundamental uestion, as we saw in the revious section, is to know the sign, ie, whether or not is a suare mod This is a very hard

More information

Complex Analysis Homework 1

Complex Analysis Homework 1 Comlex Analysis Homework 1 Steve Clanton Sarah Crimi January 27, 2009 Problem Claim. If two integers can be exressed as the sum of two squares, then so can their roduct. Proof. Call the two squares that

More information

MATH 210A, FALL 2017 HW 5 SOLUTIONS WRITTEN BY DAN DORE

MATH 210A, FALL 2017 HW 5 SOLUTIONS WRITTEN BY DAN DORE MATH 20A, FALL 207 HW 5 SOLUTIONS WRITTEN BY DAN DORE (If you find any errors, lease email ddore@stanford.edu) Question. Let R = Z[t]/(t 2 ). Regard Z as an R-module by letting t act by the identity. Comute

More information

ERRATA AND SUPPLEMENTARY MATERIAL FOR A FRIENDLY INTRODUCTION TO NUMBER THEORY FOURTH EDITION

ERRATA AND SUPPLEMENTARY MATERIAL FOR A FRIENDLY INTRODUCTION TO NUMBER THEORY FOURTH EDITION ERRATA AND SUPPLEMENTARY MATERIAL FOR A FRIENDLY INTRODUCTION TO NUMBER THEORY FOURTH EDITION JOSEPH H. SILVERMAN Acknowledgements Page vii Thanks to the following eole who have sent me comments and corrections

More information

Genus theory and the factorization of class equations over F p

Genus theory and the factorization of class equations over F p arxiv:1409.0691v2 [math.nt] 10 Dec 2017 Genus theory and the factorization of class euations over F Patrick Morton March 30, 2015 As is well-known, the Hilbert class euation is the olynomial H D (X) whose

More information

Number Theory Naoki Sato

Number Theory Naoki Sato Number Theory Naoki Sato 0 Preface This set of notes on number theory was originally written in 1995 for students at the IMO level. It covers the basic background material that an IMO

More information

arxiv: v5 [math.nt] 22 Aug 2013

arxiv: v5 [math.nt] 22 Aug 2013 Prerint, arxiv:1308900 ON SOME DETERMINANTS WITH LEGENDRE SYMBOL ENTRIES arxiv:1308900v5 [mathnt] Aug 013 Zhi-Wei Sun Deartment of Mathematics, Nanjing University Nanjing 10093, Peole s Reublic of China

More information

Small Zeros of Quadratic Forms Mod P m

Small Zeros of Quadratic Forms Mod P m International Mathematical Forum, Vol. 8, 2013, no. 8, 357-367 Small Zeros of Quadratic Forms Mod P m Ali H. Hakami Deartment of Mathematics, Faculty of Science, Jazan University P.O. Box 277, Jazan, Postal

More information

Zhi-Wei Sun Department of Mathematics, Nanjing University Nanjing , People s Republic of China

Zhi-Wei Sun Department of Mathematics, Nanjing University Nanjing , People s Republic of China Ramanuan J. 40(2016, no. 3, 511-533. CONGRUENCES INVOLVING g n (x n ( n 2 ( 2 0 x Zhi-Wei Sun Deartment of Mathematics, Naning University Naning 210093, Peole s Reublic of China zwsun@nu.edu.cn htt://math.nu.edu.cn/

More information

SQUARES IN Z/NZ. q = ( 1) (p 1)(q 1)

SQUARES IN Z/NZ. q = ( 1) (p 1)(q 1) SQUARES I Z/Z We study squares in the ring Z/Z from a theoretical and comutational oint of view. We resent two related crytograhic schemes. 1. SQUARES I Z/Z Consider for eamle the rime = 13. Write the

More information

MA3H1 Topics in Number Theory. Samir Siksek

MA3H1 Topics in Number Theory. Samir Siksek MA3H1 Toics in Number Theory Samir Siksek Samir Siksek, Mathematics Institute, University of Warwick, Coventry, CV4 7AL, United Kingdom E-mail address: samir.siksek@gmail.com Contents Chater 0. Prologue

More information

Frobenius Elements, the Chebotarev Density Theorem, and Reciprocity

Frobenius Elements, the Chebotarev Density Theorem, and Reciprocity Frobenius Elements, the Chebotarev Density Theorem, and Recirocity Dylan Yott July 30, 204 Motivation Recall Dirichlet s theorem from elementary number theory. Theorem.. For a, m) =, there are infinitely

More information

Pythagorean triples and sums of squares

Pythagorean triples and sums of squares Pythagorean triles and sums of squares Robin Chaman 16 January 2004 1 Pythagorean triles A Pythagorean trile (x, y, z) is a trile of ositive integers satisfying z 2 + y 2 = z 2. If g = gcd(x, y, z) then

More information

NUMBER SYSTEMS. Number theory is the study of the integers. We denote the set of integers by Z:

NUMBER SYSTEMS. Number theory is the study of the integers. We denote the set of integers by Z: NUMBER SYSTEMS Number theory is the study of the integers. We denote the set of integers by Z: Z = {..., 3, 2, 1, 0, 1, 2, 3,... }. The integers have two oerations defined on them, addition and multilication,

More information

PartII Number Theory

PartII Number Theory PartII Number Theory zc3 This is based on the lecture notes given by Dr.T.A.Fisher, with some other toics in number theory (ossibly not covered in the lecture). Some of the theorems here are non-examinable.

More information

Diophantine Equations

Diophantine Equations Diohantine Equations Winter Semester 018/019 University of Bayreuth Michael Stoll Contents 1. Introduction and Examles 3. Aetizers 8 3. The Law of Quadratic Recirocity 1 Print version of October 5, 018,

More information

192 VOLUME 55, NUMBER 5

192 VOLUME 55, NUMBER 5 ON THE -CLASS GROUP OF Q F WHERE F IS A PRIME FIBONACCI NUMBER MOHAMMED TAOUS Abstract Let F be a rime Fibonacci number where > Put k Q F and let k 1 be its Hilbert -class field Denote by k the Hilbert

More information

On the smallest point on a diagonal quartic threefold

On the smallest point on a diagonal quartic threefold On the smallest oint on a diagonal quartic threefold Andreas-Stehan Elsenhans and Jörg Jahnel Abstract For the family x = a y +a 2 z +a 3 v + w,,, > 0, of diagonal quartic threefolds, we study the behaviour

More information

ON THE LEAST SIGNIFICANT p ADIC DIGITS OF CERTAIN LUCAS NUMBERS

ON THE LEAST SIGNIFICANT p ADIC DIGITS OF CERTAIN LUCAS NUMBERS #A13 INTEGERS 14 (014) ON THE LEAST SIGNIFICANT ADIC DIGITS OF CERTAIN LUCAS NUMBERS Tamás Lengyel Deartment of Mathematics, Occidental College, Los Angeles, California lengyel@oxy.edu Received: 6/13/13,

More information

arxiv: v2 [math.nt] 9 Oct 2018

arxiv: v2 [math.nt] 9 Oct 2018 ON AN EXTENSION OF ZOLOTAREV S LEMMA AND SOME PERMUTATIONS LI-YUAN WANG AND HAI-LIANG WU arxiv:1810.03006v [math.nt] 9 Oct 018 Abstract. Let be an odd rime, for each integer a with a, the famous Zolotarev

More information

ANALYTIC NUMBER THEORY AND DIRICHLET S THEOREM

ANALYTIC NUMBER THEORY AND DIRICHLET S THEOREM ANALYTIC NUMBER THEORY AND DIRICHLET S THEOREM JOHN BINDER Abstract. In this aer, we rove Dirichlet s theorem that, given any air h, k with h, k) =, there are infinitely many rime numbers congruent to

More information

Chapter 3. Number Theory. Part of G12ALN. Contents

Chapter 3. Number Theory. Part of G12ALN. Contents Chater 3 Number Theory Part of G12ALN Contents 0 Review of basic concets and theorems The contents of this first section well zeroth section, really is mostly reetition of material from last year. Notations:

More information

MAS 4203 Number Theory. M. Yotov

MAS 4203 Number Theory. M. Yotov MAS 4203 Number Theory M. Yotov June 15, 2017 These Notes were comiled by the author with the intent to be used by his students as a main text for the course MAS 4203 Number Theory taught at the Deartment

More information

Characteristics of Fibonacci-type Sequences

Characteristics of Fibonacci-type Sequences Characteristics of Fibonacci-tye Sequences Yarden Blausa May 018 Abstract This aer resents an exloration of the Fibonacci sequence, as well as multi-nacci sequences and the Lucas sequence. We comare and

More information

An Overview of Witt Vectors

An Overview of Witt Vectors An Overview of Witt Vectors Daniel Finkel December 7, 2007 Abstract This aer offers a brief overview of the basics of Witt vectors. As an alication, we summarize work of Bartolo and Falcone to rove that

More information

Introduction to Arithmetic Geometry Fall 2013 Lecture #24 12/03/2013

Introduction to Arithmetic Geometry Fall 2013 Lecture #24 12/03/2013 18.78 Introduction to Arithmetic Geometry Fall 013 Lecture #4 1/03/013 4.1 Isogenies of elliptic curves Definition 4.1. Let E 1 /k and E /k be elliptic curves with distinguished rational points O 1 and

More information

p-adic Measures and Bernoulli Numbers

p-adic Measures and Bernoulli Numbers -Adic Measures and Bernoulli Numbers Adam Bowers Introduction The constants B k in the Taylor series exansion t e t = t k B k k! k=0 are known as the Bernoulli numbers. The first few are,, 6, 0, 30, 0,

More information

ON FREIMAN S 2.4-THEOREM

ON FREIMAN S 2.4-THEOREM ON FREIMAN S 2.4-THEOREM ØYSTEIN J. RØDSETH Abstract. Gregory Freiman s celebrated 2.4-Theorem says that if A is a set of residue classes modulo a rime satisfying 2A 2.4 A 3 and A < /35, then A is contained

More information

MATH 371 Class notes/outline October 15, 2013

MATH 371 Class notes/outline October 15, 2013 MATH 371 Class notes/outline October 15, 2013 More on olynomials We now consider olynomials with coefficients in rings (not just fields) other than R and C. (Our rings continue to be commutative and have

More information

On the Multiplicative Order of a n Modulo n

On the Multiplicative Order of a n Modulo n 1 2 3 47 6 23 11 Journal of Integer Sequences, Vol. 13 2010), Article 10.2.1 On the Multilicative Order of a n Modulo n Jonathan Chaelo Université Lille Nord de France F-59000 Lille France jonathan.chaelon@lma.univ-littoral.fr

More information