Math 412, Introduction to abstract algebra. Overview of algebra.
|
|
- Katherine Stevenson
- 5 years ago
- Views:
Transcription
1 Math 412, Introduction to abstract algebra. Overview of algebra. A study of algebraic objects and functions between them; an algebraic object is typically a set with one or more operations which satisfies certain axioms. Examples: vector space, eg.r 4 Main types of objects for this course: ring, eg. Z, the ring of integers having 2 operations; M n (R) in which multiplication is not commutative field, eg. Ror Q (a special type of ring; commutative and have multiplicative inverses) group, eg.z 3,S 3 ; 1 operation, commutative or not Groups have the simplest set of axioms, but are not necessarily the easiest to understand. Functions between these must preserve the operations: eg., linear transformations on vector spaces. We will have special names for the functions we wish to work with in each case (isomorphism, iso = equal, homomorphism, homo = same) To study such functions, we need to understand the kernels, special subsets which get mapped to zero (nullspaces for vector spaces, ideals for rings, normal subgroups). Chapter 2 of the book is about modular arithmetic, not because it is interesting in number theory, but because it is closely related to understanding these kernels. A topic that we will do only a little with (see grad algebra for more) is building new rings/groups out of simpler ones. For example, using Cartesian products ( 8.1 for groups). There are a number of standard methods of construction that apply to all types of algebraic objects this leads to the general idea of category theory to do them all at once. Algebraists love to generalize and find common properties in different objects. In fact, if there is one overlying theme in all of algebra, this is it: answering the question in what way are two different things the same. Historically (early in the 1900 s) abstract algebra had its beginnings with applications to algebraic geometry, algebraic topology and algebraic number theory, using algebraic ideas to help in understanding more traditional areas in which there were problems of interest to people. Thus abstract algebra played the same role in math that math has played in science, being a tool to organize the ideas. Where does high school algebra fit in? One generally studies polynomial expressions in high school. These are the elements of polynomial rings, such as R[x] ={ a i x i a i R} and will be studied in Chapters 4 and 5. This can also be generalized to polynomials in more than one variable, such as R[x, y]. As you go through this semester/year, you should be constantly asking yourselves, how 1
2 2 is what I am seeing now like something I have done before? You should always be trying to understand new definitions (of which there will be a lot) by thinking of examples you are familiar with, as well as why similar constructions are not examples. For example, Z is a commutative ring, but M n (R) isnot. Zhas a multiplicative cancellation law, but M n (R) does not. Chapter 1, Arithmetic in Z revisited. Theorem 1.1. (Division algorithm) Let a, b be integers with b>0. unique integers q, r such that Then there exist a = bq + r and 0 r<b Proof. Let S = { a bx x Z, a bx 0 } Note that S since x = a gives because b 1. a bx = a + b a = a (b ± 1) 0 Use the Well-ordering Axiom: every nonempty subset of the set of nonnegative integers contains a smallest element. (See Appendix C.) This says S has some smallest element r, say r= a bq, where qis the value of x giving r as an element of S. Wenowhavea=bq + r and r 0. We still need r<band uniqueness. Consider the integer a b(q + 1). If it is nonnegative, it lies in S; but a b(q +1)=a bq b = r b<r,
3 3 and r is the smallest element of S. Therefore, r b = a b(q +1)<0, so r<b. Uniqueness: Assume there are (possibly different) numbers q 1 and r 1 such that a = bq 1 + r 1 and 0 r 1 <b.then bq + r = a = bq 1 + r 1, so b(q q 1 )=r 1 r. This last equality can only hold for 0 r, r 1 <bif q = q 1 and r = r 1 (draw picture or see book s inequalities, p. 5). This is fundamental to the Euclidean Algorithm: Theorem 1.6. (Euclidean algorithm) Let a, b be positive integers with a b. If b a,then gcd(a, b) =b. Otherwise, apply the division algorithm repeatedly as follows; there is always an integer t such that r t is the last nonzero remainder in Then r t =gcd(a, b). a = bq 0 + r 0, 0 <r 0 <b b=r 0 q 1 +r 1, 0<r 1 <r 0 r 0 =r 1 q 2 +r 2, 0<r 2 <r 1 r t 2 =r t 1 q t +r t, r t 1 =r t q t+1 +0 To understand this, we need some definitions.. 0<r t <r t 1 Definition, p. 7. Given a, b Z, b 0,wesaybdivides a or b is a divisor or factor of a if there exists c Z such that a = bc. Notation b a. Note that, if a 0, b a,andsoahas only finitely many divisors. a =0, a =6. Examples: Definition, p. 8. common divisor; greatest common divisor (gcd). Write (a, b) or gcd(a, b). We say a and b are relatively prime if (a, b) =1. Examples: (4, 6) = 2, (5, 9) = 1. Sketch of proof of Theorem 1.6. The process stops in at most b steps since the r i s decrease. Check that r t is a common divisor and that any common divisor of a and b must divide r t.
4 4 A theorem which will be very useful later in studying ideals. Theorem 1.3. Let a, b Z, notboth0, and let d =gcd(a, b). Then there exist integers u, v with d = au + bv. Furthermore, d is the smallest positive integer that can be written in the form au + bv. Proof. The book gives a non-constructive proof, getting existence from the Well-ordering axiom. Another way is to use the Euclidean algorithm: d = r t = r t 2 r t 1 q t = r t 2 (r t 3 r t 2 q t 1 )q t = r t 2 (1 + q t 1 q t ) r t 3 = Replacing each of the r i s up to r 0 = a bq 0 gives an expression of the form au + bv where u and v are combinations of q i s. Now a = da 1 and b = db 1 for some a 1,b 1 so any positive integer of the form au + bv = d(a 1 u + b 1 v) d since a 1 u + b 1 v 1tomaketheau + bv positive. Example. Apply the Euclidean Algorithm to 114, 42 to get 6 = gcd(114, 42) = ( 8) 42. This is not unique: for example, we also have 6 = ( 4) Some standard number theory facts: Corollary 1.4 (the important part). If a, b Z are not both 0 and c a and c b, then c gcd(a, b). Proof. Write d =gcd(a, b) =au + bv = ca 1 u + cb 1 v = c(a 1 u + b 1 v). Theorem 1.5. If a bc and gcd(a, b) =1,then a c. Proof. By Theorem 1.3, we can write 1 = au + bv, so (using the hypothesis to write ar = bc), c = cau + cbv = cau + arv = a(cu + rv), and therefore a c.
5 5 Definition. An integer p is prime if p ±1 and the only divisors of p are ±1 and±p. Note that p 0. Comment on ±1 (units, p. 60). This will be generalized to arbitrary commutative rings in Chapter 6. (It can be done for noncommutative rings also.) If a number is not 0, ±1 and not prime, it is called composite. Theorem 1.8. Let p Z, p 0,±1.Thenpis prime iff p satisfies the property whenever p bc, thenp bor p c. Proof. (= ) Suppose p bc. Sincepis prime, its divisor gcd(p, b) mustbe1or p.ifitis p,then p b.if it is 1,then p cby Theorem 1.5. ( =) Let d be a divisor of p, sayp=ds. By hypothesis, p d or p s. Ifp s,says=ps 1, then p = ds = pds 1,sods 1 =1andd=±1. Otherwise, p d and d p, hence d = ±p. By definition, p is prime. Corollary 1.9. If p is prime and p a 1 a 2 a n,thenpdivides at least one of the a i. Proof. The book s proof is an informal one. To write it carefully, we need mathematical induction. The statement holds for n = 2 by Theorem 1.8. Assume it is true for n 1and n 3. Show it holds for n... Theorem 1.11 (Fundamental Theorem of Arithmetic). Every integer n 0,±1 can be written as a product of primes. The prime factorization is unique up to rearranging the factors and multiplication of the factors by ±1. Comment on proof by contradiction; i.e., use of the contrapositive to a statement. Proof. It clearly suffices to prove the theorem for n>1. Assume that some integer cannot be written as a product of primes. Let m be the smallest positive integer that cannot be so written. In particular, m is not prime, so it has positive factors other than 1 and m, saym=ab with 1 <a,b<m. Since a and b are positive integers less than m, they can be written as products of primes, say a = p 1 p r and b = q 1 q s. But then m = p 1 p r q 1 q s, a contradiction of the choice of m. Now assume that n has two prime factorizations n = p 1 p r =q 1 q s. Then p 1 divides q 1 q s, hence by Corollary 1.9, p 1 divides some q i ; reordering if necessary, we may assume i =1. Butq 1 is prime, so p 1 = ±q 1. Dividing both sides by q 1 gives us p 2 (±p 3 p r )=q 2 q s.
6 6 Use the same argument to conclude that (reordering if necessary) p 2 = ±q 2. Continue until all the primes are cancelled from one side. For example, if s>r, we end up with ±1 =q s r q s, which is impossible as the q i s are all prime. Thus we must have r = s and the factorization has the claimed uniqueness. Examples. From pages 18-19, do problems 23, 27 (generalizing part (b)), 12(a) (which leads to Chapter 2 work).
Chapter 5: The Integers
c Dr Oksana Shatalov, Fall 2014 1 Chapter 5: The Integers 5.1: Axioms and Basic Properties Operations on the set of integers, Z: addition and multiplication with the following properties: A1. Addition
More informationWORKSHEET MATH 215, FALL 15, WHYTE. We begin our course with the natural numbers:
WORKSHEET MATH 215, FALL 15, WHYTE We begin our course with the natural numbers: N = {1, 2, 3,...} which are a subset of the integers: Z = {..., 2, 1, 0, 1, 2, 3,... } We will assume familiarity with their
More informationHomework #2 Solutions Due: September 5, for all n N n 3 = n2 (n + 1) 2 4
Do the following exercises from the text: Chapter (Section 3):, 1, 17(a)-(b), 3 Prove that 1 3 + 3 + + n 3 n (n + 1) for all n N Proof The proof is by induction on n For n N, let S(n) be the statement
More informationDefinitions. Notations. Injective, Surjective and Bijective. Divides. Cartesian Product. Relations. Equivalence Relations
Page 1 Definitions Tuesday, May 8, 2018 12:23 AM Notations " " means "equals, by definition" the set of all real numbers the set of integers Denote a function from a set to a set by Denote the image of
More informationFactorization in Polynomial Rings
Factorization in Polynomial Rings Throughout these notes, F denotes a field. 1 Long division with remainder We begin with some basic definitions. Definition 1.1. Let f, g F [x]. We say that f divides g,
More information(Rgs) Rings Math 683L (Summer 2003)
(Rgs) Rings Math 683L (Summer 2003) We will first summarise the general results that we will need from the theory of rings. A unital ring, R, is a set equipped with two binary operations + and such that
More informationChapter 14: Divisibility and factorization
Chapter 14: Divisibility and factorization Matthew Macauley Department of Mathematical Sciences Clemson University http://www.math.clemson.edu/~macaule/ Math 4120, Summer I 2014 M. Macauley (Clemson) Chapter
More informationElementary Properties of the Integers
Elementary Properties of the Integers 1 1. Basis Representation Theorem (Thm 1-3) 2. Euclid s Division Lemma (Thm 2-1) 3. Greatest Common Divisor 4. Properties of Prime Numbers 5. Fundamental Theorem of
More informationNOTES ON INTEGERS. 1. Integers
NOTES ON INTEGERS STEVEN DALE CUTKOSKY The integers 1. Integers Z = {, 3, 2, 1, 0, 1, 2, 3, } have addition and multiplication which satisfy familar rules. They are ordered (m < n if m is less than n).
More informationCommutative Rings and Fields
Commutative Rings and Fields 1-22-2017 Different algebraic systems are used in linear algebra. The most important are commutative rings with identity and fields. Definition. A ring is a set R with two
More information5: The Integers (An introduction to Number Theory)
c Oksana Shatalov, Spring 2017 1 5: The Integers (An introduction to Number Theory) The Well Ordering Principle: Every nonempty subset on Z + has a smallest element; that is, if S is a nonempty subset
More informationLecture 7.4: Divisibility and factorization
Lecture 7.4: Divisibility and factorization Matthew Macauley Department of Mathematical Sciences Clemson University http://www.math.clemson.edu/~macaule/ Math 4120, Modern Algebra M. Macauley (Clemson)
More informationHandout - Algebra Review
Algebraic Geometry Instructor: Mohamed Omar Handout - Algebra Review Sept 9 Math 176 Today will be a thorough review of the algebra prerequisites we will need throughout this course. Get through as much
More information1. (a) q = 4, r = 1. (b) q = 0, r = 0. (c) q = 5, r = (a) q = 9, r = 3. (b) q = 15, r = 17. (c) q = 117, r = 11.
000 Chapter 1 Arithmetic in 1.1 The Division Algorithm Revisited 1. (a) q = 4, r = 1. (b) q = 0, r = 0. (c) q = 5, r = 3. 2. (a) q = 9, r = 3. (b) q = 15, r = 17. (c) q = 117, r = 11. 3. (a) q = 6, r =
More information32 Divisibility Theory in Integral Domains
3 Divisibility Theory in Integral Domains As we have already mentioned, the ring of integers is the prototype of integral domains. There is a divisibility relation on * : an integer b is said to be divisible
More informationMath 547, Exam 1 Information.
Math 547, Exam 1 Information. 2/10/10, LC 303B, 10:10-11:00. Exam 1 will be based on: Sections 5.1, 5.2, 5.3, 9.1; The corresponding assigned homework problems (see http://www.math.sc.edu/ boylan/sccourses/547sp10/547.html)
More informationMath 511, Algebraic Systems, Fall 2017 July 20, 2017 Edition. Todd Cochrane
Math 511, Algebraic Systems, Fall 2017 July 20, 2017 Edition Todd Cochrane Department of Mathematics Kansas State University Contents Notation v Chapter 0. Axioms for the set of Integers Z. 1 Chapter 1.
More information2 Arithmetic. 2.1 Greatest common divisors. This chapter is about properties of the integers Z = {..., 2, 1, 0, 1, 2,...}.
2 Arithmetic This chapter is about properties of the integers Z = {..., 2, 1, 0, 1, 2,...}. (See [Houston, Chapters 27 & 28]) 2.1 Greatest common divisors Definition 2.16. If a, b are integers, we say
More information1. (a) q = 4, r = 1. (b) q = 0, r = 0. (c) q = 5, r = (a) q = 9, r = 3. (b) q = 15, r = 17. (c) q = 117, r = 11.
000 Chapter 1 Arithmetic in 1.1 The Division Algorithm Revisited 1. (a) q = 4, r = 1. (b) q = 0, r = 0. (c) q = 5, r = 3. 2. (a) q = 9, r = 3. (b) q = 15, r = 17. (c) q = 117, r = 11. 3. (a) q = 6, r =
More informationCh 4.2 Divisibility Properties
Ch 4.2 Divisibility Properties - Prime numbers and composite numbers - Procedure for determining whether or not a positive integer is a prime - GCF: procedure for finding gcf (Euclidean Algorithm) - Definition:
More informationMathematical Reasoning & Proofs
Mathematical Reasoning & Proofs MAT 1362 Fall 2018 Alistair Savage Department of Mathematics and Statistics University of Ottawa This work is licensed under a Creative Commons Attribution-ShareAlike 4.0
More informationMATH FINAL EXAM REVIEW HINTS
MATH 109 - FINAL EXAM REVIEW HINTS Answer: Answer: 1. Cardinality (1) Let a < b be two real numbers and define f : (0, 1) (a, b) by f(t) = (1 t)a + tb. (a) Prove that f is a bijection. (b) Prove that any
More informationMTH 310, Section 001 Abstract Algebra I and Number Theory. Sample Midterm 1
MTH 310, Section 001 Abstract Algebra I and Number Theory Sample Midterm 1 Instructions: You have 50 minutes to complete the exam. There are five problems, worth a total of fifty points. You may not use
More informationNUMBER 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 operations defined on them, addition and multiplication,
More informationNOTES ON SIMPLE NUMBER THEORY
NOTES ON SIMPLE NUMBER THEORY DAMIEN PITMAN 1. Definitions & Theorems Definition: We say d divides m iff d is positive integer and m is an integer and there is an integer q such that m = dq. In this case,
More informationDirect Proof MAT231. Fall Transition to Higher Mathematics. MAT231 (Transition to Higher Math) Direct Proof Fall / 24
Direct Proof MAT231 Transition to Higher Mathematics Fall 2014 MAT231 (Transition to Higher Math) Direct Proof Fall 2014 1 / 24 Outline 1 Overview of Proof 2 Theorems 3 Definitions 4 Direct Proof 5 Using
More information2x 1 7. A linear congruence in modular arithmetic is an equation of the form. Why is the solution a set of integers rather than a unique integer?
Chapter 3: Theory of Modular Arithmetic 25 SECTION C Solving Linear Congruences By the end of this section you will be able to solve congruence equations determine the number of solutions find the multiplicative
More informationPRACTICE PROBLEMS: SET 1
PRACTICE PROBLEMS: SET MATH 437/537: PROF. DRAGOS GHIOCA. Problems Problem. Let a, b N. Show that if gcd(a, b) = lcm[a, b], then a = b. Problem. Let n, k N with n. Prove that (n ) (n k ) if and only if
More informationExample: This theorem is the easiest way to test an ideal (or an element) is prime. Z[x] (x)
Math 4010/5530 Factorization Theory January 2016 Let R be an integral domain. Recall that s, t R are called associates if they differ by a unit (i.e. there is some c R such that s = ct). Let R be a commutative
More informationLecture 7.5: Euclidean domains and algebraic integers
Lecture 7.5: Euclidean domains and algebraic integers Matthew Macauley Department of Mathematical Sciences Clemson University http://www.math.clemson.edu/~macaule/ Math 4120, Modern Algebra M. Macauley
More information(e) Commutativity: a b = b a. (f) Distributivity of times over plus: a (b + c) = a b + a c and (b + c) a = b a + c a.
Math 299 Midterm 2 Review Nov 4, 2013 Midterm Exam 2: Thu Nov 7, in Recitation class 5:00 6:20pm, Wells A-201. Topics 1. Methods of proof (can be combined) (a) Direct proof (b) Proof by cases (c) Proof
More informationLecture Notes Math 371: Algebra (Fall 2006) by Nathanael Leedom Ackerman
Lecture Notes Math 371: Algebra (Fall 2006) by Nathanael Leedom Ackerman October 17, 2006 TALK SLOWLY AND WRITE NEATLY!! 1 0.1 Factorization 0.1.1 Factorization of Integers and Polynomials Now we are going
More information* 8 Groups, with Appendix containing Rings and Fields.
* 8 Groups, with Appendix containing Rings and Fields Binary Operations Definition We say that is a binary operation on a set S if, and only if, a, b, a b S Implicit in this definition is the idea that
More informationMath 2070BC Term 2 Weeks 1 13 Lecture Notes
Math 2070BC 2017 18 Term 2 Weeks 1 13 Lecture Notes Keywords: group operation multiplication associative identity element inverse commutative abelian group Special Linear Group order infinite order cyclic
More information2x 1 7. A linear congruence in modular arithmetic is an equation of the form. Why is the solution a set of integers rather than a unique integer?
Chapter 3: Theory of Modular Arithmetic 25 SECTION C Solving Linear Congruences By the end of this section you will be able to solve congruence equations determine the number of solutions find the multiplicative
More informationa + bi by sending α = a + bi to a 2 + b 2. To see properties (1) and (2), it helps to think of complex numbers in polar coordinates:
5. Types of domains It turns out that in number theory the fact that certain rings have unique factorisation has very strong arithmetic consequences. We first write down some definitions. Definition 5.1.
More informationChapter 2. Divisibility. 2.1 Common Divisors
Chapter 2 Divisibility 2.1 Common Divisors Definition 2.1.1. Let a and b be integers. A common divisor of a and b is any integer that divides both a and b. Suppose that a and b are not both zero. By Proposition
More informationMath Circle Beginners Group February 28, 2016 Euclid and Prime Numbers Solutions
Math Circle Beginners Group February 28, 2016 Euclid and Prime Numbers Solutions Warm-up Problems 1. What is a prime number? Give an example of an even prime number and an odd prime number. A prime number
More information11 Division Mod n, Linear Integer Equations, Random Numbers, The Fundamental Theorem of Arithmetic
11 Division Mod n, Linear Integer Equations, Random Numbers, The Fundamental Theorem of Arithmetic Bezout s Lemma Let's look at the values of 4x + 6y when x and y are integers. If x is -6 and y is 4 we
More informationRINGS: SUMMARY OF MATERIAL
RINGS: SUMMARY OF MATERIAL BRIAN OSSERMAN This is a summary of terms used and main results proved in the subject of rings, from Chapters 11-13 of Artin. Definitions not included here may be considered
More informationSets and Motivation for Boolean algebra
SET THEORY Basic concepts Notations Subset Algebra of sets The power set Ordered pairs and Cartesian product Relations on sets Types of relations and their properties Relational matrix and the graph of
More informationNumber Theory Basics Z = {..., 2, 1, 0, 1, 2,...} For, b Z, we say that divides b if z = b for some. Notation: b Fact: for all, b, c Z:
Number Theory Basics Z = {..., 2, 1, 0, 1, 2,...} For, b Z, we say that divides b if z = b for some z Z Notation: b Fact: for all, b, c Z:, 1, and 0 0 = 0 b and b c = c b and c = (b + c) b and b = ±b 1
More informationWORKSHEET ON NUMBERS, MATH 215 FALL. We start our study of numbers with the integers: N = {1, 2, 3,...}
WORKSHEET ON NUMBERS, MATH 215 FALL 18(WHYTE) We start our study of numbers with the integers: Z = {..., 2, 1, 0, 1, 2, 3,... } and their subset of natural numbers: N = {1, 2, 3,...} For now we will not
More informationProofs. Chapter 2 P P Q Q
Chapter Proofs In this chapter we develop three methods for proving a statement. To start let s suppose the statement is of the form P Q or if P, then Q. Direct: This method typically starts with P. Then,
More informationGroups, Rings, and Finite Fields. Andreas Klappenecker. September 12, 2002
Background on Groups, Rings, and Finite Fields Andreas Klappenecker September 12, 2002 A thorough understanding of the Agrawal, Kayal, and Saxena primality test requires some tools from algebra and elementary
More informationMATH10040 Chapter 1: Integers and divisibility
MATH10040 Chapter 1: Integers and divisibility Recall the basic definition: 1. Divisibilty Definition 1.1. If a, b Z, we say that b divides a, or that a is a multiple of b and we write b a if there is
More informationChapter 9, Additional topics for integral domains
Chapter 9, Additional topics for integral domains Many times we have mentioned that theorems we proved could be done much more generally they only required some special property like unique factorization,
More informationChapter 3. Rings. The basic commutative rings in mathematics are the integers Z, the. Examples
Chapter 3 Rings Rings are additive abelian groups with a second operation called multiplication. The connection between the two operations is provided by the distributive law. Assuming the results of Chapter
More informationDiscrete valuation rings. Suppose F is a field. A discrete valuation on F is a function v : F {0} Z such that:
Discrete valuation rings Suppose F is a field. A discrete valuation on F is a function v : F {0} Z such that: 1. v is surjective. 2. v(ab) = v(a) + v(b). 3. v(a + b) min(v(a), v(b)) if a + b 0. Proposition:
More informationContribution of Problems
Exam topics 1. Basic structures: sets, lists, functions (a) Sets { }: write all elements, or define by condition (b) Set operations: A B, A B, A\B, A c (c) Lists ( ): Cartesian product A B (d) Functions
More informationNumber Theory Proof Portfolio
Number Theory Proof Portfolio Jordan Rock May 12, 2015 This portfolio is a collection of Number Theory proofs and problems done by Jordan Rock in the Spring of 2014. The problems are organized first by
More informationDivisibility = 16, = 9, = 2, = 5. (Negative!)
Divisibility 1-17-2018 You probably know that division can be defined in terms of multiplication. If m and n are integers, m divides n if n = mk for some integer k. In this section, I ll look at properties
More information2 Elementary number theory
2 Elementary number theory 2.1 Introduction Elementary number theory is concerned with properties of the integers. Hence we shall be interested in the following sets: The set if integers {... 2, 1,0,1,2,3,...},
More informationNOTES ON FINITE FIELDS
NOTES ON FINITE FIELDS AARON LANDESMAN CONTENTS 1. Introduction to finite fields 2 2. Definition and constructions of fields 3 2.1. The definition of a field 3 2.2. Constructing field extensions by adjoining
More information8 Primes and Modular Arithmetic
8 Primes and Modular Arithmetic 8.1 Primes and Factors Over two millennia ago already, people all over the world were considering the properties of numbers. One of the simplest concepts is prime numbers.
More informationModular Arithmetic Instructor: Marizza Bailey Name:
Modular Arithmetic Instructor: Marizza Bailey Name: 1. Introduction to Modular Arithmetic If someone asks you what day it is 145 days from now, what would you answer? Would you count 145 days, or find
More informationChapter 1 A Survey of Divisibility 14
Chapter 1 A Survey of Divisibility 14 SECTION C Euclidean Algorithm By the end of this section you will be able to use properties of the greatest common divisor (gcd) obtain the gcd using the Euclidean
More informationALGEBRA HANDOUT 2.3: FACTORIZATION IN INTEGRAL DOMAINS. In this handout we wish to describe some aspects of the theory of factorization.
ALGEBRA HANDOUT 2.3: FACTORIZATION IN INTEGRAL DOMAINS PETE L. CLARK In this handout we wish to describe some aspects of the theory of factorization. The first goal is to state what it means for an arbitrary
More informationRing Theory Problem Set 2 Solutions
Ring Theory Problem Set 2 Solutions 16.24. SOLUTION: We already proved in class that Z[i] is a commutative ring with unity. It is the smallest subring of C containing Z and i. If r = a + bi is in Z[i],
More information4 Powers of an Element; Cyclic Groups
4 Powers of an Element; Cyclic Groups Notation When considering an abstract group (G, ), we will often simplify notation as follows x y will be expressed as xy (x y) z will be expressed as xyz x (y z)
More informationAbstract algebra is the study of structures that certain collections of. 204, linear algebra, or in CS 221, discrete structures.
Math 375 Week 1 1.1 Introduction to Groups Introduction Abstract algebra is the study of structures that certain collections of `objects' or `sets' possess. You have already had a taste of this in Math
More informationContribution of Problems
Exam topics 1. Basic structures: sets, lists, functions (a) Sets { }: write all elements, or define by condition (b) Set operations: A B, A B, A\B, A c (c) Lists ( ): Cartesian product A B (d) Functions
More informationRings If R is a commutative ring, a zero divisor is a nonzero element x such that xy = 0 for some nonzero element y R.
Rings 10-26-2008 A ring is an abelian group R with binary operation + ( addition ), together with a second binary operation ( multiplication ). Multiplication must be associative, and must distribute over
More informationINTEGERS. In this section we aim to show the following: Goal. Every natural number can be written uniquely as a product of primes.
INTEGERS PETER MAYR (MATH 2001, CU BOULDER) In this section we aim to show the following: Goal. Every natural number can be written uniquely as a product of primes. 1. Divisibility Definition. Let a, b
More information1. Introduction to commutative rings and fields
1. Introduction to commutative rings and fields Very informally speaking, a commutative ring is a set in which we can add, subtract and multiply elements so that the usual laws hold. A field is a commutative
More informationFall 2017 Test II review problems
Fall 2017 Test II review problems Dr. Holmes October 18, 2017 This is a quite miscellaneous grab bag of relevant problems from old tests. Some are certainly repeated. 1. Give the complete addition and
More informationRings and Fields Theorems
Rings and Fields Theorems Rajesh Kumar PMATH 334 Intro to Rings and Fields Fall 2009 October 25, 2009 12 Rings and Fields 12.1 Definition Groups and Abelian Groups Let R be a non-empty set. Let + and (multiplication)
More informationContents. 4 Arithmetic and Unique Factorization in Integral Domains. 4.1 Euclidean Domains and Principal Ideal Domains
Ring Theory (part 4): Arithmetic and Unique Factorization in Integral Domains (by Evan Dummit, 018, v. 1.00) Contents 4 Arithmetic and Unique Factorization in Integral Domains 1 4.1 Euclidean Domains and
More informationWriting Assignment 2 Student Sample Questions
Writing Assignment 2 Student Sample Questions 1. Let P and Q be statements. Then the statement (P = Q) ( P Q) is a tautology. 2. The statement If the sun rises from the west, then I ll get out of the bed.
More informationGreatest Common Divisor MATH Greatest Common Divisor. Benjamin V.C. Collins, James A. Swenson MATH 2730
MATH 2730 Greatest Common Divisor Benjamin V.C. Collins James A. Swenson The world s least necessary definition Definition Let a, b Z, not both zero. The largest integer d such that d a and d b is called
More information1. multiplication is commutative and associative;
Chapter 4 The Arithmetic of Z In this chapter, we start by introducing the concept of congruences; these are used in our proof (going back to Gauss 1 ) that every integer has a unique prime factorization.
More information1 Overview and revision
MTH6128 Number Theory Notes 1 Spring 2018 1 Overview and revision In this section we will meet some of the concerns of Number Theory, and have a brief revision of some of the relevant material from Introduction
More informationNotes for Math 290 using Introduction to Mathematical Proofs by Charles E. Roberts, Jr.
Notes for Math 290 using Introduction to Mathematical Proofs by Charles E. Roberts, Jr. Chapter : Logic Topics:. Statements, Negation, and Compound Statements.2 Truth Tables and Logical Equivalences.3
More informationBasic Algebra. Final Version, August, 2006 For Publication by Birkhäuser Boston Along with a Companion Volume Advanced Algebra In the Series
Basic Algebra Final Version, August, 2006 For Publication by Birkhäuser Boston Along with a Companion Volume Advanced Algebra In the Series Cornerstones Selected Pages from Chapter I: pp. 1 15 Anthony
More informationWritten Homework # 4 Solution
Math 516 Fall 2006 Radford Written Homework # 4 Solution 12/10/06 You may use results form the book in Chapters 1 6 of the text, from notes found on our course web page, and results of the previous homework.
More informationMATH 2200 Final Review
MATH 00 Final Review Thomas Goller December 7, 01 1 Exam Format The final exam will consist of 8-10 proofs It will take place on Tuesday, December 11, from 10:30 AM - 1:30 PM, in the usual room Topics
More informationChapter 1. Greatest common divisor. 1.1 The division theorem. In the beginning, there are the natural numbers 0, 1, 2, 3, 4,...,
Chapter 1 Greatest common divisor 1.1 The division theorem In the beginning, there are the natural numbers 0, 1, 2, 3, 4,..., which constitute the set N. Addition and multiplication are binary operations
More informationCHAPTER 4: EXPLORING Z
CHAPTER 4: EXPLORING Z MATH 378, CSUSM. SPRING 2009. AITKEN 1. Introduction In this chapter we continue the study of the ring Z. We begin with absolute values. The absolute value function Z N is the identity
More informationThe Euclidean Algorithm and Multiplicative Inverses
1 The Euclidean Algorithm and Multiplicative Inverses Lecture notes for Access 2009 The Euclidean Algorithm is a set of instructions for finding the greatest common divisor of any two positive integers.
More informationMath 547, Exam 2 Information.
Math 547, Exam 2 Information. 3/19/10, LC 303B, 10:10-11:00. Exam 2 will be based on: Homework and textbook sections covered by lectures 2/3-3/5. (see http://www.math.sc.edu/ boylan/sccourses/547sp10/547.html)
More informationCoding Theory ( Mathematical Background I)
N.L.Manev, Lectures on Coding Theory (Maths I) p. 1/18 Coding Theory ( Mathematical Background I) Lector: Nikolai L. Manev Institute of Mathematics and Informatics, Sofia, Bulgaria N.L.Manev, Lectures
More informationOlympiad Number Theory Through Challenging Problems
Olympiad Number Theory Justin Stevens Page 1 Olympiad Number Theory Through Challenging Problems Authors Justin Stevens Editor and L A TEX Manager David Altizio Dedicated to my sister. Justin Contents
More informationMAT246H1S - Concepts In Abstract Mathematics. Solutions to Term Test 1 - February 1, 2018
MAT246H1S - Concepts In Abstract Mathematics Solutions to Term Test 1 - February 1, 2018 Time allotted: 110 minutes. Aids permitted: None. Comments: Statements of Definitions, Principles or Theorems should
More informationPrime and irreducible elements of the ring of integers modulo n
Prime and irreducible elements of the ring of integers modulo n M. H. Jafari and A. R. Madadi Department of Pure Mathematics, Faculty of Mathematical Sciences University of Tabriz, Tabriz, Iran Abstract
More information1. Introduction to commutative rings and fields
1. Introduction to commutative rings and fields Very informally speaking, a commutative ring is a set in which we can add, subtract and multiply elements so that the usual laws hold. A field is a commutative
More informationPrime Rational Functions and Integral Polynomials. Jesse Larone, Bachelor of Science. Mathematics and Statistics
Prime Rational Functions and Integral Polynomials Jesse Larone, Bachelor of Science Mathematics and Statistics Submitted in partial fulfillment of the requirements for the degree of Master of Science Faculty
More informationMODEL ANSWERS TO HWK #10
MODEL ANSWERS TO HWK #10 1. (i) As x + 4 has degree one, either it divides x 3 6x + 7 or these two polynomials are coprime. But if x + 4 divides x 3 6x + 7 then x = 4 is a root of x 3 6x + 7, which it
More informationCS 5319 Advanced Discrete Structure. Lecture 9: Introduction to Number Theory II
CS 5319 Advanced Discrete Structure Lecture 9: Introduction to Number Theory II Divisibility Outline Greatest Common Divisor Fundamental Theorem of Arithmetic Modular Arithmetic Euler Phi Function RSA
More informationChapter 5. Number Theory. 5.1 Base b representations
Chapter 5 Number Theory The material in this chapter offers a small glimpse of why a lot of facts that you ve probably nown and used for a long time are true. It also offers some exposure to generalization,
More informationRings and modular arithmetic
Chapter 8 Rings and modular arithmetic So far, we have been working with just one operation at a time. But standard number systems, such as Z, have two operations + and which interact. It is useful to
More informationIF A PRIME DIVIDES A PRODUCT... ζ(s) = n s. ; p s
IF A PRIME DIVIDES A PRODUCT... STEVEN J. MILLER AND CESAR E. SILVA ABSTRACT. One of the greatest difficulties encountered by all in their first proof intensive class is subtly assuming an unproven fact
More informationLecture 7: Polynomial rings
Lecture 7: Polynomial rings Rajat Mittal IIT Kanpur You have seen polynomials many a times till now. The purpose of this lecture is to give a formal treatment to constructing polynomials and the rules
More informationBeautiful Mathematics
Beautiful Mathematics 1. Principle of Mathematical Induction The set of natural numbers is the set of positive integers {1, 2, 3,... } and is denoted by N. The Principle of Mathematical Induction is a
More informationSolutions to odd-numbered exercises Peter J. Cameron, Introduction to Algebra, Chapter 2
Solutions to odd-numbered exercises Peter J Cameron, Introduction to Algebra, Chapter 1 The answers are a No; b No; c Yes; d Yes; e No; f Yes; g Yes; h No; i Yes; j No a No: The inverse law for addition
More informationPRIME NUMBERS YANKI LEKILI
PRIME NUMBERS YANKI LEKILI We denote by N the set of natural numbers: 1,2,..., These are constructed using Peano axioms. We will not get into the philosophical questions related to this and simply assume
More informationCHAPTER 8: EXPLORING R
CHAPTER 8: EXPLORING R LECTURE NOTES FOR MATH 378 (CSUSM, SPRING 2009). WAYNE AITKEN In the previous chapter we discussed the need for a complete ordered field. The field Q is not complete, so we constructed
More information1. Factorization Divisibility in Z.
8 J. E. CREMONA 1.1. Divisibility in Z. 1. Factorization Definition 1.1.1. Let a, b Z. Then we say that a divides b and write a b if b = ac for some c Z: a b c Z : b = ac. Alternatively, we may say that
More informationALGEBRA. 1. Some elementary number theory 1.1. Primes and divisibility. We denote the collection of integers
ALGEBRA CHRISTIAN REMLING 1. Some elementary number theory 1.1. Primes and divisibility. We denote the collection of integers by Z = {..., 2, 1, 0, 1,...}. Given a, b Z, we write a b if b = ac for some
More informationNotes for Math 345. Dan Singer Minnesota State University, Mankato. August 23, 2006
Notes for Math 345 Dan Singer Minnesota State University, Mankato August 23, 2006 Preliminaries 1. Read the To The Student section pp. xvi-xvii and the Thematic Table of Contents. 2. Read Appendix A: Logic
More informationAlgebraic structures I
MTH5100 Assignment 1-10 Algebraic structures I For handing in on various dates January March 2011 1 FUNCTIONS. Say which of the following rules successfully define functions, giving reasons. For each one
More information