# Ideal Class Group and Units

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 Chapter 4 Ideal Class Group and Units We are now interested in understanding two aspects of ring of integers of number fields: how principal they are (that is, what is the proportion of principal ideals among all the ideals), and what is the structure of their group of units For the former task, we will introduce the notion of class number (as the measure of how principal a ring of integers is), and prove that the class number is finite We will then prove Dirichlet s Theorem for the structure of groups of units Both results will be derived in the spirit of geometry of numbers, that is as a consequence of Minkowski s theorem, where algebraic results are proved thanks to a suitable geometrical interpretation (mainly the fact that a ring of integers can be seen as a lattice in R n via the n embeddings of its number field) 41 Ideal class group Let K be a number field, and O K be its ring of integers We have seen in Chapter 2 that we can extend the notion of ideal to fractional ideal, and that with this new notion, we have a group structure (Theorem 25) Let I K denote the group of fractional ideals of K Let P K denote the subgroup of I K formed by the principal ideals, that is ideals of the form αo K, α K Definition 41 The ideal class group, denoted by Cl(K), is Cl(K) = I K /P K Definition 42 We denote by h K the cardinality Cl K, called the class number In particular, if O K is a principal ideal domain, then Cl(K) = 0, and h K = 1 Our goal is now to prove that the class number is finite for ring of integers of number fields The lemma below is a version of Minkowski s theorem 45

2 46 CHAPTER 4 IDEAL CLASS GROUP AND UNITS Lemma 41 Let Λ be a lattice of R n Let X R n be a convex, compact set (that is a closed and bounded set since we are in R n ), which is symmetric with respect to 0 (that is, x X x X) If Vol(X) 2 n Vol(R n /Λ), then there exists 0 λ Λ such that λ X Proof Let us first assume that the inequality is strict: Vol(X) > 2 n Vol(R n /Λ) Let us consider the map ψ : 1 2 X = {x 2 Rn x X} R n /Λ If ψ were injective, then ( ) 1 Vol 2 X = 1 2 n Vol(X) Vol(Rn /Λ) that is Vol(X) 2 n Vol(R n /Λ), which contradicts our assumption Thus ψ cannot be injective, which means that there exist x 1 x 2 1 2X such that ψ(x 1 ) = ψ(x 2 ) By symmetry, we have that x 2 1 2X, and by convexity of X (that is (1 t)x + ty X for t [0,1]), we have that ( 1 1 ) x ( x 2) = x 1 x X Thus 0 λ = x 1 x 2 X, and λ Λ (since ψ(x 1 x 2 ) = 0) Let us now assume that Vol(X) = 2 n Vol(R n /Λ) By what we have just proved, there exists 0 λ ǫ Λ such that λ ǫ (1 + ǫ)x for all ǫ > 0, since Vol((1 + ǫ)x) = (1 + ǫ) n Vol(X) = (1 + ǫ) n 2 n Vol(R n /Λ) > 2 n Vol(R n /Λ), for all ǫ > 0 In particular, if ǫ < 1, then λ ǫ 2X Λ The set 2X Λ is compact and discrete (since Λ is discrete), it is thus finite Let us now understand what is happening here On the one hand, we have a sequence λ ǫ with infinitely many terms since there is one for every 0 < ǫ < 1, while on the other hand, those infinitely many terms are all lattice points in 2X, which only contains finitely many of them This means that this sequence must converge to a point 0 λ Λ which belongs to (1 + ǫ)x for infinitely many ǫ > 0 Thus λ Λ ( ǫ 0 (1 + ǫ)x 0) Since X is closed, we have that λ X Let n = [K : Q] be the degree of K and let (r 1,r 2 ) be the signature of K Let σ 1,,σ r1 be the r 1 real embeddings of K into R We choose one of the two embeddings in each pair of complex embeddings, which we denote by

3 41 IDEAL CLASS GROUP 47 σ r1+1,,σ r1+r 2 We consider the following map, called canonical embedding of K: σ : K R r1 C r2 R n α (σ 1 (α),,σ r1 (α),σ r1+1(α),,σ r1+r 2 (α)) (41) We have that the image of O K by σ is a lattice σ(o K ) in R n (we have that σ(o K ) is a free abelian group, which contains a basis of R n ) Let α 1,,α n be a Z-basis of O K Let M be the generator matrix of the lattice σ(o K ), given by σ 1 (α 1 ) σ r1 (α 1 ) Re(σ r1+1(α 1 )) Im(σ r1+1(α 1 )) Re(σ r1+r 2 (α 1 )) Im(σ r1+r 2 (α 1 )) σ 1 (α n ) σ r1 (α n ) Re(σ r1+1(α n )) Im(σ r1+1(α n )) Re(σ r1+r 2 (α n )) Im(σ r1+r 2 (α n )) whose determinant is given by Vol(R n /σ(o K )) = det(m) = K 2 r2 Indeed, we have that Re(x) = (x + x)/2 and Im(x) = (x x)/2i, x C, and det(m) = det(m ) where M is given by σ 1 (α 1 ) σ r1 (α 1 ) σ r1+1(α 1 ) σ 1 (α n ) σ r1 (α n ) σ r1+1(α n ) σ r1 +1(α 1) σ r1 +1(α 1) 2i σ r1+r 2 (α 1 ) σ r1 +1(α n) σ r1 +1(α n) 2i σ r1+r 2 (α n ) Again, we have that det(m ) = 2 r2 det(m ), with M given by this time σ 1 (α 1 ) σ r1 (α 1 ) σ r1+1(α 1 ) σ r1+1(α 1 ) σ r1+r 2 (α 1 ) σ r1+r 2 (α 1 ), σ 1 (α n ) σ r1 (α n ) σ r1+1(α n ) σ r1+1(α n ) σ r1+r 2 (α n ) σ r1+r 2 (α n ) σ r1 +r 2 (α 1) σ r1 +r 2 (α 1) 2i σ r1 +r 2 (α n) σ r1 +r 2 (α n) 2i which concludes the proof, since (recall that complex embeddings come by pairs of conjugates) det(m) = 2 r2 det(m ) = 2 r2 K We are now ready to prove that Cl(K) = I K /P K is finite Theorem 42 Let K be a number field with discriminant K 1 There exists a constant C = C r1,r 2 > 0 (which only depends on r 1 and r 2 ) such that every ideal class (that is every coset of Cl(K)) contains an integral ideal whose norm is at most C K

4 48 CHAPTER 4 IDEAL CLASS GROUP AND UNITS 2 The group Cl(K) is finite Proof Recall first that by definition, a non-zero fractional ideal J is a finitely generated O K -submodule of K, and there exists β K such that βj O K (if β i span J as O K -module, write β i = δ i /γ i and set β = γ i ) The fact that βj O K exactly means that β J 1 by definition of the inverse of a fractional ideal (see Chapter 2) The idea of the proof consists of, given a fractional ideal J, looking at the norm of a corresponding integral ideal βj, which we will prove is bounded as claimed Let us pick a non-zero fractional ideal I Since I is a finitely generated O K - module, we have that σ(i) is a lattice in R n, and so is σ(i 1 ), with the property that Vol(R n /σ(i 1 )) = Vol(R n /σ(o K ))N(I 1 K ) = 2 r2 N(I), where the first equality comes from the fact that the volume is given by the determinant of the generator matrix of the lattice Now since we have two lattices, we can write the generator matrix of σ(i 1 ) as being the generator matrix of σ(o K ) multiplied by a matrix whose determinant in absolute value is the index of the two lattices Let X be a compact convex set, symmetrical with respect to 0 In order to get a set of volume big enough to use Minkowski theorem, we set a scaling factor so that the volume of λx is λ n = 2 n Vol(Rn /σ(i 1 )), Vol(X) Vol(λX) = λ n Vol(X) = 2 n Vol(R n /σ(i 1 )) By Lemma 41, there exists 0 σ(α) σ(i 1 ) and σ(α) λx Since α I 1, we have that αi is an integral ideal in the same ideal class as I, and N(αI) = N K/Q (α) N(I) = n σ i (α) N(I) Mλ n N(I), where M = max x X xi, x = (x 1,,x n ), so that the maximum over λx gives λ n M Thus, by definition of λ n, we have that N(αI) 2n Vol(R n /σ(i 1 )) MN(I) Vol(X) 2 n M = K 2 r2 Vol(X) M = 2r1+r2 K Vol(X) } {{ } C This completes the first part of the proof

5 42 DIRICHLET UNITS THEOREM 49 Figure 41: Johann Peter Gustav Lejeune Dirichlet ( ) We are now left to prove that Cl(K) is a finite group By what we have just proved, we can find a system of representatives J i of I K /P K consisting of integral ideals J i, of norm smaller than C K In particular, the prime factors of J i have a norm smaller than C K Above the prime numbers p < C K, there are only finitely many prime ideals (or in other words, there are only finitely many integrals with a given norm) 42 Dirichlet Units Theorem By abuse of language, we call units of K the units of O K, that is the invertible elements of O K We have seen early on (Corollary 111) that units are characterized by their norm, namely units are exactly the elements of O K with norm ±1 Theorem 43 (Dirichlet Units Theorem) Let K be a number field of degree n, with signature (r 1,r 2 ) The group OK of units of K is the product of the group µ(o K ) of roots of unity in O K, which is cyclic and finite, and a free group on r 1 + r 2 1 generators In formula, we have that O K Z r1+r2 1 µ(o K ) The most difficult part of this theorem is actually to prove that the free group has exactly r 1 + r 2 1 generators This is nowadays usually proven using Minkowski s theorem Dirichlet though did not have Minkowski s theorem available: he proved the unit theorem in 1846 while Minkowski developed the geometry of numbers only around the end of the 19th century He used instead the pigeonhole principle It is said that Dirichlet got the main idea for his proof while attending a concert in the Sistine Chapel

6 50 CHAPTER 4 IDEAL CLASS GROUP AND UNITS Proof Let σ : K R r1 C r2 R n be the canonical embedding of K (see (41)) The logarithmic embedding of K is the mapping λ : K R r1+r2 α (log σ 1 (α),,log σ r1+r 2 (α)) Since λ(αβ) = λ(α)+λ(β), λ is a homomorphism from the multiplicative group K to the additive group of R r1+r2 Step 1 We first prove that the kernel of λ restricted to OK is a finite group In order to do so, we prove that if C is a bounded subset of R r1+r2, then C = {x OK, λ(x) C} is a finite set In words, we look at the preimage of a bounded set by the logarithmic embedding (more precisely, at the restriction of the preimage to the units of O K ) Proof Since C is bounded, all σ i (x), x OK, i = 1,,n belong to some interval say [a 1,a], a > 1 Thus the elementary polynomials in the σ i (x) will also belong to some interval of the same form Now they are the coefficients of the characteristic polynomial of x, which has integer coefficients since x OK Thus there are only finitely many possible characteristic polynomials of elements x C, hence only finitely many possible roots of minimal polynomials of elements x C, which shows that x can belong to C for only finitely many x Now if we set C = {0}, C is the kernel ker(λ) O K of λ restricted to OK and is thus finite Step 2 We now show that ker(λ) O K consists of exactly all the roots of unity µ(o K ) Proof That it does consist of roots of unity (and is cyclic) is a known property of any subgroup of the multiplicative group of any field Thus if x ker(λ) O K then x is a root of unity Now conversely, suppose that x m = 1 Then x is an algebraic integer, and σ i (x) m = σ i (x m ) = 1 = 1 so that σ i (x) = 1, and thus log σ i (x) = 0 for all i, showing that x ker(λ) O K Step 3 We are now ready to prove that OK is a finite generated abelian group, isomorphic to µ(o K ) Z s, s r 1 + r 2 Proof By Step 1, we know that λ(ok ) is a discrete subgroup of, that is, Rr1+r2 any bounded subset of R r1+r2 contains only finitely many points of λ(ok ) Thus λ(ok ) is a lattice in Rs, hence a free Z-module of rank s, for some s r 1 + r 2 Now by the first isomorphism theorem, we have that λ(o K) O K/µ(O K ) with λ(x) corresponding to the coset xµ(o K ) If x 1 µ(o K ),,x s µ(o K ) form a basis for OK /µ(o K) and x OK, then xµ(o K) is a finite produce of powers of the x i G, so x is an element of µ(o K ) times a finite product of powers of the x i Since the λ(x i ) are linearly independent, so are the x i (provided that the notion of linear independence is translated to a multiplicative setting: x 1,,x s are multiplicatively independent if x m1 1 x ms s = 1 implies that m i = 0 for all i,

7 42 DIRICHLET UNITS THEOREM 51 from which it follows that x m1 1 x ms s = x n1 1 xns s implies m i = n i for all i) The result follows Step 4 We now improve the estimate of s and show that s r 1 + r 2 1 Proof If x is a unit, then we know that its norm must be ±1 Then ±1 = N(x) = n σ i (x) = r 1 σ i (x) r 1+r 2 j=r 1+1 σ j (x)σ j (x) By taking the absolute values and applying the logarithmic embedding, we get 0 = r 1 log σ i (x) + r 1+r 2 j=r 1+1 log( σ j (x) σ j (x) ) and λ(x) = (y 1,,y r1+r 2 ) lies in the hyperplane W whose equation is r 1 y i + 2 r 1+r 2 j=r 1+1 y j = 0 The hyperplane has dimension r 1 +r 2 1, so as above, λ(ok ) is a free Z-module of rank s r 1 + r 2 1 Step 5 We are left with showing that s = r 1 +r 2 1, which is actually the hardest part of the proof This uses Minkowski theorem The proof may come later one proof can be found in the online lecture of Robert Ash Example 41 Consider K an imaginary quadratic field, that is of the form K = Q( d), with d a positive square free integer Its signature is (r 1,r 2 ) = (0,1) We thus have that its group of units is given by Z r1+r2 1 G = G, that is only roots of unity Actually, we have that the units are the 4rth roots of unity if K = Q( 1) (that is ±1, ±i), the 6th roots of unity if K = Q( 3) (that is ±1, ±ζ 3, ±ζ 2 3), and only ±1 otherwise Example 42 For K = Q( 3), we have (r 1,r 2 ) = (2,0), thus r 1 + r 2 1 = 1, and µ(o K ) = ±1 The unit group is given by O K ±(2 + 3) Z The main definitions and results of this chapter are Definition of ideal class group and class number The fact that the class number of a number field is finite The structure of units in a number field (the statement of Dirichlet s theorem)

8 52 CHAPTER 4 IDEAL CLASS GROUP AND UNITS

### The Dirichlet Unit Theorem

Chapter 6 The Dirichlet Unit Theorem As usual, we will be working in the ring B of algebraic integers of a number field L. Two factorizations of an element of B are regarded as essentially the same if

### The Ideal Class Group

Chapter 5 The Ideal Class Group We will use Minkowski theory, which belongs to the general area of geometry of numbers, to gain insight into the ideal class group of a number field. We have already mentioned

### A number field is a field of finite degree over Q. By the Primitive Element Theorem, any number

Number Fields Introduction A number field is a field of finite degree over Q. By the Primitive Element Theorem, any number field K = Q(α) for some α K. The minimal polynomial Let K be a number field and

### U.C. Berkeley CS276: Cryptography Handout 0.1 Luca Trevisan January, 2009. Notes on Algebra

U.C. Berkeley CS276: Cryptography Handout 0.1 Luca Trevisan January, 2009 Notes on Algebra These notes contain as little theory as possible, and most results are stated without proof. Any introductory

### Mathematics Course 111: Algebra I Part IV: Vector Spaces

Mathematics Course 111: Algebra I Part IV: Vector Spaces D. R. Wilkins Academic Year 1996-7 9 Vector Spaces A vector space over some field K is an algebraic structure consisting of a set V on which are

### 7. Some irreducible polynomials

7. Some irreducible polynomials 7.1 Irreducibles over a finite field 7.2 Worked examples Linear factors x α of a polynomial P (x) with coefficients in a field k correspond precisely to roots α k [1] of

### I. GROUPS: BASIC DEFINITIONS AND EXAMPLES

I GROUPS: BASIC DEFINITIONS AND EXAMPLES Definition 1: An operation on a set G is a function : G G G Definition 2: A group is a set G which is equipped with an operation and a special element e G, called

### OSTROWSKI FOR NUMBER FIELDS

OSTROWSKI FOR NUMBER FIELDS KEITH CONRAD Ostrowski classified the nontrivial absolute values on Q: up to equivalence, they are the usual (archimedean) absolute value and the p-adic absolute values for

### Factoring of Prime Ideals in Extensions

Chapter 4 Factoring of Prime Ideals in Extensions 4. Lifting of Prime Ideals Recall the basic AKLB setup: A is a Dedekind domain with fraction field K, L is a finite, separable extension of K of degree

### IRREDUCIBLE OPERATOR SEMIGROUPS SUCH THAT AB AND BA ARE PROPORTIONAL. 1. Introduction

IRREDUCIBLE OPERATOR SEMIGROUPS SUCH THAT AB AND BA ARE PROPORTIONAL R. DRNOVŠEK, T. KOŠIR Dedicated to Prof. Heydar Radjavi on the occasion of his seventieth birthday. Abstract. Let S be an irreducible

### EXERCISES FOR THE COURSE MATH 570, FALL 2010

EXERCISES FOR THE COURSE MATH 570, FALL 2010 EYAL Z. GOREN (1) Let G be a group and H Z(G) a subgroup such that G/H is cyclic. Prove that G is abelian. Conclude that every group of order p 2 (p a prime

### MA257: INTRODUCTION TO NUMBER THEORY LECTURE NOTES

MA257: INTRODUCTION TO NUMBER THEORY LECTURE NOTES 2016 47 4. Diophantine Equations A Diophantine Equation is simply an equation in one or more variables for which integer (or sometimes rational) solutions

### Introduction to Finite Fields (cont.)

Chapter 6 Introduction to Finite Fields (cont.) 6.1 Recall Theorem. Z m is a field m is a prime number. Theorem (Subfield Isomorphic to Z p ). Every finite field has the order of a power of a prime number

### Introduction to finite fields

Introduction to finite fields Topics in Finite Fields (Fall 2013) Rutgers University Swastik Kopparty Last modified: Monday 16 th September, 2013 Welcome to the course on finite fields! This is aimed at

### Continued Fractions and the Euclidean Algorithm

Continued Fractions and the Euclidean Algorithm Lecture notes prepared for MATH 326, Spring 997 Department of Mathematics and Statistics University at Albany William F Hammond Table of Contents Introduction

### it is easy to see that α = a

21. Polynomial rings Let us now turn out attention to determining the prime elements of a polynomial ring, where the coefficient ring is a field. We already know that such a polynomial ring is a UF. Therefore

### 11 Ideals. 11.1 Revisiting Z

11 Ideals The presentation here is somewhat different than the text. In particular, the sections do not match up. We have seen issues with the failure of unique factorization already, e.g., Z[ 5] = O Q(

### Cyclotomic Extensions

Chapter 7 Cyclotomic Extensions A cyclotomic extension Q(ζ n ) of the rationals is formed by adjoining a primitive n th root of unity ζ n. In this chapter, we will find an integral basis and calculate

### Numerical Analysis Lecture Notes

Numerical Analysis Lecture Notes Peter J. Olver 5. Inner Products and Norms The norm of a vector is a measure of its size. Besides the familiar Euclidean norm based on the dot product, there are a number

### (x + a) n = x n + a Z n [x]. Proof. If n is prime then the map

22. A quick primality test Prime numbers are one of the most basic objects in mathematics and one of the most basic questions is to decide which numbers are prime (a clearly related problem is to find

### ADDITIVE GROUPS OF RINGS WITH IDENTITY

ADDITIVE GROUPS OF RINGS WITH IDENTITY SIMION BREAZ AND GRIGORE CĂLUGĂREANU Abstract. A ring with identity exists on a torsion Abelian group exactly when the group is bounded. The additive groups of torsion-free

### On the representability of the bi-uniform matroid

On the representability of the bi-uniform matroid Simeon Ball, Carles Padró, Zsuzsa Weiner and Chaoping Xing August 3, 2012 Abstract Every bi-uniform matroid is representable over all sufficiently large

### GROUPS SUBGROUPS. Definition 1: An operation on a set G is a function : G G G.

Definition 1: GROUPS An operation on a set G is a function : G G G. Definition 2: A group is a set G which is equipped with an operation and a special element e G, called the identity, such that (i) the

### FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 22

FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 22 RAVI VAKIL CONTENTS 1. Discrete valuation rings: Dimension 1 Noetherian regular local rings 1 Last day, we discussed the Zariski tangent space, and saw that it

### Chapter 13: Basic ring theory

Chapter 3: Basic ring theory Matthew Macauley Department of Mathematical Sciences Clemson University http://www.math.clemson.edu/~macaule/ Math 42, Spring 24 M. Macauley (Clemson) Chapter 3: Basic ring

### The Open University s repository of research publications and other research outputs

Open Research Online The Open University s repository of research publications and other research outputs The degree-diameter problem for circulant graphs of degree 8 and 9 Journal Article How to cite:

### Monogenic Fields and Power Bases Michael Decker 12/07/07

Monogenic Fields and Power Bases Michael Decker 12/07/07 1 Introduction Let K be a number field of degree k and O K its ring of integers Then considering O K as a Z-module, the nicest possible case is

### Unique Factorization

Unique Factorization Waffle Mathcamp 2010 Throughout these notes, all rings will be assumed to be commutative. 1 Factorization in domains: definitions and examples In this class, we will study the phenomenon

### Metric Spaces. Chapter 7. 7.1. Metrics

Chapter 7 Metric Spaces A metric space is a set X that has a notion of the distance d(x, y) between every pair of points x, y X. The purpose of this chapter is to introduce metric spaces and give some

### FUNCTIONAL ANALYSIS LECTURE NOTES: QUOTIENT SPACES

FUNCTIONAL ANALYSIS LECTURE NOTES: QUOTIENT SPACES CHRISTOPHER HEIL 1. Cosets and the Quotient Space Any vector space is an abelian group under the operation of vector addition. So, if you are have studied

### GROUP ALGEBRAS. ANDREI YAFAEV

GROUP ALGEBRAS. ANDREI YAFAEV We will associate a certain algebra to a finite group and prove that it is semisimple. Then we will apply Wedderburn s theory to its study. Definition 0.1. Let G be a finite

### 1 = (a 0 + b 0 α) 2 + + (a m 1 + b m 1 α) 2. for certain elements a 0,..., a m 1, b 0,..., b m 1 of F. Multiplying out, we obtain

Notes on real-closed fields These notes develop the algebraic background needed to understand the model theory of real-closed fields. To understand these notes, a standard graduate course in algebra is

### CARDINALITY, COUNTABLE AND UNCOUNTABLE SETS PART ONE

CARDINALITY, COUNTABLE AND UNCOUNTABLE SETS PART ONE With the notion of bijection at hand, it is easy to formalize the idea that two finite sets have the same number of elements: we just need to verify

### ZORN S LEMMA AND SOME APPLICATIONS

ZORN S LEMMA AND SOME APPLICATIONS KEITH CONRAD 1. Introduction Zorn s lemma is a result in set theory that appears in proofs of some non-constructive existence theorems throughout mathematics. We will

### Prime Numbers and Irreducible Polynomials

Prime Numbers and Irreducible Polynomials M. Ram Murty The similarity between prime numbers and irreducible polynomials has been a dominant theme in the development of number theory and algebraic geometry.

### some algebra prelim solutions

some algebra prelim solutions David Morawski August 19, 2012 Problem (Spring 2008, #5). Show that f(x) = x p x + a is irreducible over F p whenever a F p is not zero. Proof. First, note that f(x) has no

### THE FUNDAMENTAL THEOREM OF ALGEBRA VIA PROPER MAPS

THE FUNDAMENTAL THEOREM OF ALGEBRA VIA PROPER MAPS KEITH CONRAD 1. Introduction The Fundamental Theorem of Algebra says every nonconstant polynomial with complex coefficients can be factored into linear

### MOP 2007 Black Group Integer Polynomials Yufei Zhao. Integer Polynomials. June 29, 2007 Yufei Zhao yufeiz@mit.edu

Integer Polynomials June 9, 007 Yufei Zhao yufeiz@mit.edu We will use Z[x] to denote the ring of polynomials with integer coefficients. We begin by summarizing some of the common approaches used in dealing

### Algebra 1 Course Title

Algebra 1 Course Title Course- wide 1. What patterns and methods are being used? Course- wide 1. Students will be adept at solving and graphing linear and quadratic equations 2. Students will be adept

### G = G 0 > G 1 > > G k = {e}

Proposition 49. 1. A group G is nilpotent if and only if G appears as an element of its upper central series. 2. If G is nilpotent, then the upper central series and the lower central series have the same

### SUBGROUPS OF CYCLIC GROUPS. 1. Introduction In a group G, we denote the (cyclic) group of powers of some g G by

SUBGROUPS OF CYCLIC GROUPS KEITH CONRAD 1. Introduction In a group G, we denote the (cyclic) group of powers of some g G by g = {g k : k Z}. If G = g, then G itself is cyclic, with g as a generator. Examples

### ISU Department of Mathematics. Graduate Examination Policies and Procedures

ISU Department of Mathematics Graduate Examination Policies and Procedures There are four primary criteria to be used in evaluating competence on written or oral exams. 1. Knowledge Has the student demonstrated

### Introduction to Modern Algebra

Introduction to Modern Algebra David Joyce Clark University Version 0.0.6, 3 Oct 2008 1 1 Copyright (C) 2008. ii I dedicate this book to my friend and colleague Arthur Chou. Arthur encouraged me to write

### SMALL SKEW FIELDS CÉDRIC MILLIET

SMALL SKEW FIELDS CÉDRIC MILLIET Abstract A division ring of positive characteristic with countably many pure types is a field Wedderburn showed in 1905 that finite fields are commutative As for infinite

### 4.1 Modules, Homomorphisms, and Exact Sequences

Chapter 4 Modules We always assume that R is a ring with unity 1 R. 4.1 Modules, Homomorphisms, and Exact Sequences A fundamental example of groups is the symmetric group S Ω on a set Ω. By Cayley s Theorem,

### BookTOC.txt. 1. Functions, Graphs, and Models. Algebra Toolbox. Sets. The Real Numbers. Inequalities and Intervals on the Real Number Line

College Algebra in Context with Applications for the Managerial, Life, and Social Sciences, 3rd Edition Ronald J. Harshbarger, University of South Carolina - Beaufort Lisa S. Yocco, Georgia Southern University

### Mathematics Review for MS Finance Students

Mathematics Review for MS Finance Students Anthony M. Marino Department of Finance and Business Economics Marshall School of Business Lecture 1: Introductory Material Sets The Real Number System Functions,

### CONTINUED FRACTIONS AND FACTORING. Niels Lauritzen

CONTINUED FRACTIONS AND FACTORING Niels Lauritzen ii NIELS LAURITZEN DEPARTMENT OF MATHEMATICAL SCIENCES UNIVERSITY OF AARHUS, DENMARK EMAIL: niels@imf.au.dk URL: http://home.imf.au.dk/niels/ Contents

### COMMUTATIVITY DEGREES OF WREATH PRODUCTS OF FINITE ABELIAN GROUPS

COMMUTATIVITY DEGREES OF WREATH PRODUCTS OF FINITE ABELIAN GROUPS IGOR V. EROVENKO AND B. SURY ABSTRACT. We compute commutativity degrees of wreath products A B of finite abelian groups A and B. When B

### Chapter 4, Arithmetic in F [x] Polynomial arithmetic and the division algorithm.

Chapter 4, Arithmetic in F [x] Polynomial arithmetic and the division algorithm. We begin by defining the ring of polynomials with coefficients in a ring R. After some preliminary results, we specialize

### Subsets of Euclidean domains possessing a unique division algorithm

Subsets of Euclidean domains possessing a unique division algorithm Andrew D. Lewis 2009/03/16 Abstract Subsets of a Euclidean domain are characterised with the following objectives: (1) ensuring uniqueness

### Galois Theory III. 3.1. Splitting fields.

Galois Theory III. 3.1. Splitting fields. We know how to construct a field extension L of a given field K where a given irreducible polynomial P (X) K[X] has a root. We need a field extension of K where

### s-convexity, model sets and their relation

s-convexity, model sets and their relation Zuzana Masáková Jiří Patera Edita Pelantová CRM-2639 November 1999 Department of Mathematics, Faculty of Nuclear Science and Physical Engineering, Czech Technical

### The van Hoeij Algorithm for Factoring Polynomials

The van Hoeij Algorithm for Factoring Polynomials Jürgen Klüners Abstract In this survey we report about a new algorithm for factoring polynomials due to Mark van Hoeij. The main idea is that the combinatorial

### BANACH AND HILBERT SPACE REVIEW

BANACH AND HILBET SPACE EVIEW CHISTOPHE HEIL These notes will briefly review some basic concepts related to the theory of Banach and Hilbert spaces. We are not trying to give a complete development, but

### POLYNOMIAL RINGS AND UNIQUE FACTORIZATION DOMAINS

POLYNOMIAL RINGS AND UNIQUE FACTORIZATION DOMAINS RUSS WOODROOFE 1. Unique Factorization Domains Throughout the following, we think of R as sitting inside R[x] as the constant polynomials (of degree 0).

### First and raw version 0.1 23. september 2013 klokken 13:45

The discriminant First and raw version 0.1 23. september 2013 klokken 13:45 One of the most significant invariant of an algebraic number field is the discriminant. One is tempted to say, apart from the

### Week 5 Integral Polyhedra

Week 5 Integral Polyhedra We have seen some examples 1 of linear programming formulation that are integral, meaning that every basic feasible solution is an integral vector. This week we develop a theory

### Proof. The map. G n i. where d is the degree of D.

7. Divisors Definition 7.1. We say that a scheme X is regular in codimension one if every local ring of dimension one is regular, that is, the quotient m/m 2 is one dimensional, where m is the unique maximal

### 3 1. Note that all cubes solve it; therefore, there are no more

Math 13 Problem set 5 Artin 11.4.7 Factor the following polynomials into irreducible factors in Q[x]: (a) x 3 3x (b) x 3 3x + (c) x 9 6x 6 + 9x 3 3 Solution: The first two polynomials are cubics, so if

### Group Theory. Contents

Group Theory Contents Chapter 1: Review... 2 Chapter 2: Permutation Groups and Group Actions... 3 Orbits and Transitivity... 6 Specific Actions The Right regular and coset actions... 8 The Conjugation

### CONTRIBUTIONS TO ZERO SUM PROBLEMS

CONTRIBUTIONS TO ZERO SUM PROBLEMS S. D. ADHIKARI, Y. G. CHEN, J. B. FRIEDLANDER, S. V. KONYAGIN AND F. PAPPALARDI Abstract. A prototype of zero sum theorems, the well known theorem of Erdős, Ginzburg

### 1 Sets and Set Notation.

LINEAR ALGEBRA MATH 27.6 SPRING 23 (COHEN) LECTURE NOTES Sets and Set Notation. Definition (Naive Definition of a Set). A set is any collection of objects, called the elements of that set. We will most

PYTHAGOREAN TRIPLES KEITH CONRAD 1. Introduction A Pythagorean triple is a triple of positive integers (a, b, c) where a + b = c. Examples include (3, 4, 5), (5, 1, 13), and (8, 15, 17). Below is an ancient

### 1 Local Brouwer degree

1 Local Brouwer degree Let D R n be an open set and f : S R n be continuous, D S and c R n. Suppose that the set f 1 (c) D is compact. (1) Then the local Brouwer degree of f at c in the set D is defined.

### minimal polyonomial Example

Minimal Polynomials Definition Let α be an element in GF(p e ). We call the monic polynomial of smallest degree which has coefficients in GF(p) and α as a root, the minimal polyonomial of α. Example: We

### University of Lille I PC first year list of exercises n 7. Review

University of Lille I PC first year list of exercises n 7 Review Exercise Solve the following systems in 4 different ways (by substitution, by the Gauss method, by inverting the matrix of coefficients

### by the matrix A results in a vector which is a reflection of the given

Eigenvalues & Eigenvectors Example Suppose Then So, geometrically, multiplying a vector in by the matrix A results in a vector which is a reflection of the given vector about the y-axis We observe that

### Module MA3411: Abstract Algebra Galois Theory Appendix Michaelmas Term 2013

Module MA3411: Abstract Algebra Galois Theory Appendix Michaelmas Term 2013 D. R. Wilkins Copyright c David R. Wilkins 1997 2013 Contents A Cyclotomic Polynomials 79 A.1 Minimum Polynomials of Roots of

### THE DIMENSION OF A VECTOR SPACE

THE DIMENSION OF A VECTOR SPACE KEITH CONRAD This handout is a supplementary discussion leading up to the definition of dimension and some of its basic properties. Let V be a vector space over a field

### The cover SU(2) SO(3) and related topics

The cover SU(2) SO(3) and related topics Iordan Ganev December 2011 Abstract The subgroup U of unit quaternions is isomorphic to SU(2) and is a double cover of SO(3). This allows a simple computation of

### Linear Algebra I. Ronald van Luijk, 2012

Linear Algebra I Ronald van Luijk, 2012 With many parts from Linear Algebra I by Michael Stoll, 2007 Contents 1. Vector spaces 3 1.1. Examples 3 1.2. Fields 4 1.3. The field of complex numbers. 6 1.4.

### CONTINUED FRACTIONS, PELL S EQUATION, AND TRANSCENDENTAL NUMBERS

CONTINUED FRACTIONS, PELL S EQUATION, AND TRANSCENDENTAL NUMBERS JEREMY BOOHER Continued fractions usually get short-changed at PROMYS, but they are interesting in their own right and useful in other areas

### Four Unsolved Problems in Congruence Permutable Varieties

Four Unsolved Problems in Congruence Permutable Varieties Ross Willard University of Waterloo, Canada Nashville, June 2007 Ross Willard (Waterloo) Four Unsolved Problems Nashville, June 2007 1 / 27 Congruence

### ALGEBRA HW 5 CLAY SHONKWILER

ALGEBRA HW 5 CLAY SHONKWILER 510.5 Let F = Q(i). Prove that x 3 and x 3 3 are irreducible over F. Proof. If x 3 is reducible over F then, since it is a polynomial of degree 3, it must reduce into a product

### 1 if 1 x 0 1 if 0 x 1

Chapter 3 Continuity In this chapter we begin by defining the fundamental notion of continuity for real valued functions of a single real variable. When trying to decide whether a given function is or

### MA651 Topology. Lecture 6. Separation Axioms.

MA651 Topology. Lecture 6. Separation Axioms. This text is based on the following books: Fundamental concepts of topology by Peter O Neil Elements of Mathematics: General Topology by Nicolas Bourbaki Counterexamples

### = 2 + 1 2 2 = 3 4, Now assume that P (k) is true for some fixed k 2. This means that

Instructions. Answer each of the questions on your own paper, and be sure to show your work so that partial credit can be adequately assessed. Credit will not be given for answers (even correct ones) without

### GROUPS ACTING ON A SET

GROUPS ACTING ON A SET MATH 435 SPRING 2012 NOTES FROM FEBRUARY 27TH, 2012 1. Left group actions Definition 1.1. Suppose that G is a group and S is a set. A left (group) action of G on S is a rule for

### Revision of ring theory

CHAPTER 1 Revision of ring theory 1.1. Basic definitions and examples In this chapter we will revise and extend some of the results on rings that you have studied on previous courses. A ring is an algebraic

### 16.3 Fredholm Operators

Lectures 16 and 17 16.3 Fredholm Operators A nice way to think about compact operators is to show that set of compact operators is the closure of the set of finite rank operator in operator norm. In this

### Research Article On the Rank of Elliptic Curves in Elementary Cubic Extensions

Numbers Volume 2015, Article ID 501629, 4 pages http://dx.doi.org/10.1155/2015/501629 Research Article On the Rank of Elliptic Curves in Elementary Cubic Extensions Rintaro Kozuma College of International

### Galois representations with open image

Galois representations with open image Ralph Greenberg University of Washington Seattle, Washington, USA May 7th, 2011 Introduction This talk will be about representations of the absolute Galois group

### ON GENERALIZED RELATIVE COMMUTATIVITY DEGREE OF A FINITE GROUP. A. K. Das and R. K. Nath

International Electronic Journal of Algebra Volume 7 (2010) 140-151 ON GENERALIZED RELATIVE COMMUTATIVITY DEGREE OF A FINITE GROUP A. K. Das and R. K. Nath Received: 12 October 2009; Revised: 15 December

### Let H and J be as in the above lemma. The result of the lemma shows that the integral

Let and be as in the above lemma. The result of the lemma shows that the integral ( f(x, y)dy) dx is well defined; we denote it by f(x, y)dydx. By symmetry, also the integral ( f(x, y)dx) dy is well defined;

### The determinant of a skew-symmetric matrix is a square. This can be seen in small cases by direct calculation: 0 a. 12 a. a 13 a 24 a 14 a 23 a 14

4 Symplectic groups In this and the next two sections, we begin the study of the groups preserving reflexive sesquilinear forms or quadratic forms. We begin with the symplectic groups, associated with

### Elements of Abstract Group Theory

Chapter 2 Elements of Abstract Group Theory Mathematics is a game played according to certain simple rules with meaningless marks on paper. David Hilbert The importance of symmetry in physics, and for

### Lecture 11. Shuanglin Shao. October 2nd and 7th, 2013

Lecture 11 Shuanglin Shao October 2nd and 7th, 2013 Matrix determinants: addition. Determinants: multiplication. Adjoint of a matrix. Cramer s rule to solve a linear system. Recall that from the previous

### THE FUNDAMENTAL THEOREM OF ARBITRAGE PRICING

THE FUNDAMENTAL THEOREM OF ARBITRAGE PRICING 1. Introduction The Black-Scholes theory, which is the main subject of this course and its sequel, is based on the Efficient Market Hypothesis, that arbitrages

### UNIVERSITY GRADUATE STUDIES PROGRAM SILLIMAN UNIVERSITY DUMAGUETE CITY. Master of Science in Mathematics

1 UNIVERSITY GRADUATE STUDIES PROGRAM SILLIMAN UNIVERSITY DUMAGUETE CITY Master of Science in Mathematics Introduction The Master of Science in Mathematics (MS Math) program is intended for students who

### 4: SINGLE-PERIOD MARKET MODELS

4: SINGLE-PERIOD MARKET MODELS Ben Goldys and Marek Rutkowski School of Mathematics and Statistics University of Sydney Semester 2, 2015 B. Goldys and M. Rutkowski (USydney) Slides 4: Single-Period Market

### MATH10212 Linear Algebra. Systems of Linear Equations. Definition. An n-dimensional vector is a row or a column of n numbers (or letters): a 1.

MATH10212 Linear Algebra Textbook: D. Poole, Linear Algebra: A Modern Introduction. Thompson, 2006. ISBN 0-534-40596-7. Systems of Linear Equations Definition. An n-dimensional vector is a row or a column

### Row Ideals and Fibers of Morphisms

Michigan Math. J. 57 (2008) Row Ideals and Fibers of Morphisms David Eisenbud & Bernd Ulrich Affectionately dedicated to Mel Hochster, who has been an inspiration to us for many years, on the occasion

### Zeros of Polynomial Functions

Review: Synthetic Division Find (x 2-5x - 5x 3 + x 4 ) (5 + x). Factor Theorem Solve 2x 3-5x 2 + x + 2 =0 given that 2 is a zero of f(x) = 2x 3-5x 2 + x + 2. Zeros of Polynomial Functions Introduction

### PUTNAM TRAINING POLYNOMIALS. Exercises 1. Find a polynomial with integral coefficients whose zeros include 2 + 5.

PUTNAM TRAINING POLYNOMIALS (Last updated: November 17, 2015) Remark. This is a list of exercises on polynomials. Miguel A. Lerma Exercises 1. Find a polynomial with integral coefficients whose zeros include

### 1 VECTOR SPACES AND SUBSPACES

1 VECTOR SPACES AND SUBSPACES What is a vector? Many are familiar with the concept of a vector as: Something which has magnitude and direction. an ordered pair or triple. a description for quantities such

### Reading material on the limit set of a Fuchsian group

Reading material on the limit set of a Fuchsian group Recommended texts Many books on hyperbolic geometry and Kleinian and Fuchsian groups contain material about limit sets. The presentation given here

### Some Polynomial Theorems. John Kennedy Mathematics Department Santa Monica College 1900 Pico Blvd. Santa Monica, CA 90405 rkennedy@ix.netcom.

Some Polynomial Theorems by John Kennedy Mathematics Department Santa Monica College 1900 Pico Blvd. Santa Monica, CA 90405 rkennedy@ix.netcom.com This paper contains a collection of 31 theorems, lemmas,