Chapter 3, Rings. Definitions and examples.


 Kory Wilkinson
 2 years ago
 Views:
Transcription
1 Chapter 3, Rings Definitions and examples. We now have several examples of algebraic systems with addition and multiplication: Z, Z n, R, M n (R), 2Z = {2n n Z}. We will write down a system of axioms which includes them all. Definition, p. 42. A ring is a nonempty set R with two binary operations (usually written as addition and multiplication) such that for all a, b, c R, (1) R is closed under addition: a + b R. (2) Addition is associative: (a + b)+c=a+(b+c). (3) Addition is commutative: a + b = b + a. (4) R contains an additive identity element, called zero and usually denoted by 0 or 0 R : a +0=0+a=a. (5) Every element of R has an additive inverse: for each a, there exists an x R such that a + x =0=x+a. We write x = a. (6) R is closed under multiplication: ab R. (7) Multiplication is associative: (ab)c = a(bc). (8) Multiplication distributes over addition: a(b + c) =ab + ac and (a + b)c = ac + bc. Other possible properties are captured by special types of rings. We will encounter many in this book; many more are studied as well by mathematicians. Theorem 3.3. Additive inverses are unique. Proof. Assume that x and y are both inverses of a. Then x = x +0 = x+(a+y)= (x+a)+y=0+y=y. We don t have to add axioms about subtraction. We just define a b to be a +( b). Definition. A commutative ring is a ring R that satisfies the additional axiom that ab = ba for all a, b R. Examples are Z, R, Z n,2z, but not M n (R) ifn 2. Definition. A ring with identity is a ring R that contains a multiplicative identity element 1 R :1 R a=a=a1 R for all a R. ( ) 1 0 Examples: 1 in the first three rings above, in M (R). The set of even integers 1
2 2 is a ring without identity. The set of odd integers is not a ring. We can also work with matrices whose elements come from any ring we know about, such as M n (Z r ). Example. Let ( R = ) M 2 (Z 2 ). This is a finite ((16 elements) ) noncommutative ring with identity 1 R = and zero element R =. We give an example to show it 0 0 is noncommutative: ( )( ) ( ) ( )( ) ( ) = but = Example: rings of continuous functions. Let X be any topological space; if you don t know what that is, let it be R or any interval in R. We consider the set R = C(X, R), the set of all continuous functions from X to R. R becomes a ring with identity when we define addition and multiplication as in elementary calculus: (f + g)(x) =f(x)+g(x)and (fg)(x) = f(x)g(x). The identity element is the constant function 1. R is commutative because R is, but it does have zero divisors for almost all choices of X. There are many, many examples of this sort of ring. The functions don t have to be continuous. They can be restricted in many other ways, or not restricted at all. For example, you can look at polynomial functions or differentiable functions (for some choices of X). Definition, p. 46. An integral domain is a commutative ring R with identity 1 R 0 R with no zero divisors; that is, ab =0 R implies that a =0 R or b =0 R. Examples: Z, R, Z p for p prime. Nonexamples: C(R, R), Z n for n composite, the zero ring {0 R }, the even integers 2Z. Definition, p. 47. A field is an integral domain in which every nonzero element a has a multiplicative inverse, denoted a 1. Examples: R, Q, C, Z p for p prime (Theorem 2.8). If an element of a ring has a multiplicative inverse, it is unique. The proof is the same as that given above for Theorem 3.3 if we replace addition by multiplication. (Note that we did not use the commutativity of addition.) This is also the proof from Math 311 that invertible matrices have unique inverses. Definition, p. 60. Any element a in a ring R with identity which has an inverse u (i.e., au =1 r =ua) iscalledaunit.
3 3 Making new rings. Theorem 3.1 (Product rings). Let R, S be rings and form the Cartesian product R S. Define operations by (r, s)+(r,s )=(r+r,s+s ) (r, s)(r,s )=(rr,ss ). Then R S is a ring. If R and S are both commutative, so is R S. If R and S both have an identity, then (1 R, 1 S ) is the identity in R S. Example. Let R be the ring Z Z = { (n, m) n, m Z }. Note that (1,0)(0, 1) = (0, 0) = 0 R,soRis not an integral domain. For the same reason, no product ring is an integral domain. Definition, p. 49. A subset of a ring which is itself a ring (using the same operations) is called a subring. A subset of a field which is itself a field is called a subfield. Q is a subfield of R, and both are subfields of C. Z is a subring of Q. Z 3 is not a subring of Z. Its elements are not integers, but rather are congruence classes of integers. 2Z = { 2n n Z } is a subring of Z, but the only subring of Z with identity is Z itself. The zero ring is a subring of every ring. As with subspaces of vector spaces, it is not hard to check that a subset is a subring as most axioms are inherited from the ring. Theorem 3.2. Let S be a subset of a ring R. S is a subring of R iff the following conditions all hold: (1) S is closed under addition and multiplication. (2) 0 R S. (3) s S for every element s S. Proof. Axioms 1, 4, 5 and 6 are in our hypotheses. Axioms 2, 3, 7 and 8 are inherited from R. As examples, do exercises from book, page 51 55: 7, 8, 9, 19, 34, discuss 35.
4 4 Basic properties of rings. Theorem For all a, b, c in a ring R, (1) a + b = a + c implies b = c. (2) a 0=0 a=0. (3) a( b) = (ab) =( a)b. (4) ( a) =a. (5) (a + b) =( a)+( b). (6) ( a)( b) =ab. (7) ( 1)a = a if R has an identity element. Proof. These make use of the definition of subtraction and negatives. For (1), add a. (2) a 0=a(0 + 0) = a 0+a 0anduse(1). (3) Show that a( b) and( a)bare additive inverses for ab (we know there is a unique such element). (4) and (5) Definition of additive inverse. (6) Use (3) and (4). (7) Use (3) or add ( 1)a to a. A quicker version of Theorem 3.2 can be obtained using subtraction. Theorem 3.6. Let R be a ring and S R. S is a subring of R if S is closed under subtraction and multiplication. Proof. We need to show S is closed under addition, has 0 and has additive inverses. But S implies there is some s S, hence 0 = s s S. For any a S, a =0 a S. And finally, for a, b S, a + b = a ( b) S. Notation: For a in a ring R and n Z, write na for a sum of n copies of a and a n for a product of n copies of a. This includes the cases 0 a =0 R and a 0 =1 R if R has an identity element. Be careful: (a+b) 2 = a 2 +ab+ba+b 2 cannot be simplified unless the ring is commutative. Theorem 3.7. Let R be a ring and a, b R. The equation a + x = b has the unique solution x = b a in R. Proof. b a is a solution: check it! If z is another solution, then a + z = b = a +(b a), so z = b a by cancellation (Theorem 3.4). For multiplication, we need multiplicative inverses to get the cancellation in the proof.
5 Theorem 3.8. Let R be a ring with identity and a, b R. Ifais a unit, then the equations ax = b and ya = b have unique solutions in R. 5 Proof. x = a 1 b and y = ba 1 are solutions: check! Uniqueness works as in Theorem 3.7, using the inverse for cancellation: if z is another solution to ax = b, thenaz = b = a(a 1 b). Multiply on the left by a 1 to get z = a 1 az = a 1 a(a 1 b)=a 1 b. A similar argument works for y. The solutions x = a 1 b and y = ba 1 may not be the same. Exercise 4, p. 62 gives an example with 2 2 matrices. Sometimes multiplicative cancellation works without inverses (recall the integers). Theorem Let R be an integral domain. If a 0and ab = ac, then b =c. Proof. ab = ac implies a(b c) = 0. Since a 0andRis an integral domain, we must have b c =0,orb=c. If R is not an integral domain, this fails because of zero divisors: p. 62. Let a 0 in a ring R. a is a zero divisor if there exists an element b 0inR with either ab =0orba =0. Example: 2 3=0=2 0inZ 6. Theorem Every finite integral domain is a field. Combinatorial proof. Let a 0 in the integral domain R. The set ar = { ar r R } is a permutation of the elements of R: ax = ay implies x = y by Theorem 3.10 and there are only finitely many elements. Therefore some ar must be 1 and a has an inverse. Examples. Discuss exercises 8, 11, 17, 25 on pages
6 6 Isomorphisms and homomorphisms. Recall from linear algebra that a linear transformation is a function between vector spaces that preserves the operations on the vector space. For rings we only want to consider the functions that preserve their two operations. Definition, p. 71. Let R and S be rings. A function f : R S (which means the domain is R and f takes values in S) iscalledahomomorphism if f(a + b) =f(a)+f(b)and f(ab) =f(a)f(b) for all a, b R. Examples. Z Q defined by n n is the natural embedding of the integers into the rational numbers. Z Z n defined by n [n]. This is a homomorphism by the definition of addition and multiplication in Z n. (Theorem 2.6) ι R : R R, the identity map for any ring R. f : C C defined by f(a + bi) =a bi (complex conjugation). Check the definition. The last two examples are special in that they are onetoone (injective) andonto (surjective). In this case, we say the homomorphism is an isomorphism. Iff: R Sis a ring isomorphism, we say R and S are isomorphic. An interesting example in the book is given on p. ( 69: the ) complex numbers are isomorphic to the ring of real 2 2 matrices a b of the form, which corresponds to the complex number a + bi. b a Besides the identity, there is one other rather trivial example: the zero mapping z : R S defined by z(r) =0 S for all r R is a ring homomorphism. Example. Recall Exer. 7, p. 51: R = { (r, r) r R }. Define f : R R by f (r) =(r, r), Check that f is an isomorphism. Theorem Let f : R S be a ring homomorphism. (1) f(0 R )=0 S. (2) f( a) = f(a)for all a R. (3) f(a b) =f(a) f(b)for all a, b R. If R is a ring with identity and f is surjective, then (4) S is a ring with identity and f(1 R )=1 S. (5) Whenever u R is a unit, then f(u) is a unit in S and f(u) 1 = f(u 1 ). Proof. (1) f(0) = f(0 + 0) = f(0) + f(0) implies f(0 R )=0 S. (2) f( a)+f(a)=f( a+a)=f(0) = 0 S by (1), so f( a) = f(a). (3) f(a b) =f(a+( b)) = f(a)+f( b)=f(a) f(b)by(2).
7 (4) Let s S. We must show sf(1) = f(1)s = s. Sincefis surjective, there is some r R with f(r) =s.thensf(1) = f(r)f(1) = f(r 1) = f(r) =sand similarly for f(1)s. (5) f(u 1 )f(u) =f(u 1 u)=f(1) = 1 by (4). Note that we really need surjectivity in (4) and (5). If S = R R and we define f : R S by f (r) =(r, 0), then f(1) = (1, 0) is not the identity in S. Furthermore, 1 R is a unit, but f(1) is not a unit. When the homomorphism f : R S is not surjective, it is handy to have a name for the subset of S that it maps onto. We write im(f) ={f(r) r R}and call this set the image of f. (The book does not have a symbol for the image or for the kernel which we shall define later.) Now we can say that f : R S is surjective if S =im(f). 7 Corollary Given a ring homomorphism f : R S, the image of f is a subring of S. Proof. im(f) since f(0) = 0 im(f). Closure under subtraction is Theorem 3.12(3). Closure under multiplication follows from the definition of homomorphism: f(a)f(b) = f(ab) im(f). Students should read pages for hints on how to decide (or prove) whether two rings are isomorphic or not. Examples. 1. Z 6 is isomorphic to Z 2 Z 3. They have the same number of elements. We must find a bijection which preserves the operations. Define f : Z 6 Z 2 Z 3 by f ([n]) = ([n] 2, [n] 3 ). Check that f is a homomorphism. Since both sets have 6 elements, f will be a bijection if it is either injective OR surjective. Surjectivity is an example of the Chinese Remainder Theorem (page 408). We show injectivity. Assume f([n]) = f([m]). Then f([m n]) = f([m]) f([n]) = 0, so it suffices to prove only [0] maps to ([0], [0]). (Recall the situation for vector spaces!) If [a] ([0] 2, [0] 3 ), then a is divisible by both 2 and 3, hence by 6 since they are relatively prime. Therefore [a] =0inZ Z 4 is not isomorphic to Z 2 Z 2. What fails in trying to use the previous proof is that 2 and 2 are not relatively prime. What is really different about them? Any isomorphism must take 1 Z 4 to (1, 1) Z 2 Z 2.ButinZ 4, we must add 1 to itself four times to get zero, while (1, 1) + (1, 1) = (0, 0). Thus any homomorphism f : Z 4 Z 2 Z 2 will have f(2) = f(0) = 0 and will not be injective. These two examples can be generalized. See exercises 39, 40 on page Pages 76 79, exercises 18, 25, 33.
Elementary Number Theory We begin with a bit of elementary number theory, which is concerned
CONSTRUCTION OF THE FINITE FIELDS Z p S. R. DOTY Elementary Number Theory We begin with a bit of elementary number theory, which is concerned solely with questions about the set of integers Z = {0, ±1,
More informationChapter 10. Abstract algebra
Chapter 10. Abstract algebra C.O.S. Sorzano Biomedical Engineering December 17, 2013 10. Abstract algebra December 17, 2013 1 / 62 Outline 10 Abstract algebra Sets Relations and functions Partitions and
More informationCOMMUTATIVE RINGS. Definition: A domain is a commutative ring R that satisfies the cancellation law for multiplication:
COMMUTATIVE RINGS Definition: A commutative ring R is a set with two operations, addition and multiplication, such that: (i) R is an abelian group under addition; (ii) ab = ba for all a, b R (commutative
More informationMathematics Course 111: Algebra I Part IV: Vector Spaces
Mathematics Course 111: Algebra I Part IV: Vector Spaces D. R. Wilkins Academic Year 19967 9 Vector Spaces A vector space over some field K is an algebraic structure consisting of a set V on which are
More informationChapter 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
More information5.1 Commutative rings; Integral Domains
5.1 J.A.Beachy 1 5.1 Commutative rings; Integral Domains from A Study Guide for Beginner s by J.A.Beachy, a supplement to Abstract Algebra by Beachy / Blair 23. Let R be a commutative ring. Prove the following
More informationSo let us begin our quest to find the holy grail of real analysis.
1 Section 5.2 The Complete Ordered Field: Purpose of Section We present an axiomatic description of the real numbers as a complete ordered field. The axioms which describe the arithmetic of the real numbers
More informationChapter 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
More informationAbstract Algebra Cheat Sheet
Abstract Algebra Cheat Sheet 16 December 2002 By Brendan Kidwell, based on Dr. Ward Heilman s notes for his Abstract Algebra class. Notes: Where applicable, page numbers are listed in parentheses at the
More informationI. 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
More informationNotes on Algebraic Structures. Peter J. Cameron
Notes on Algebraic Structures Peter J. Cameron ii Preface These are the notes of the secondyear course Algebraic Structures I at Queen Mary, University of London, as I taught it in the second semester
More informationFinite Fields and ErrorCorrecting Codes
Lecture Notes in Mathematics Finite Fields and ErrorCorrecting Codes KarlGustav Andersson (Lund University) (version 1.01316 September 2015) Translated from Swedish by Sigmundur Gudmundsson Contents
More information3. Equivalence Relations. Discussion
3. EQUIVALENCE RELATIONS 33 3. Equivalence Relations 3.1. Definition of an Equivalence Relations. Definition 3.1.1. A relation R on a set A is an equivalence relation if and only if R is reflexive, symmetric,
More informationArkansas Tech University MATH 4033: Elementary Modern Algebra Dr. Marcel B. Finan
Arkansas Tech University MATH 4033: Elementary Modern Algebra Dr. Marcel B. Finan 3 Binary Operations We are used to addition and multiplication of real numbers. These operations combine two real numbers
More informationUnique 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
More informationCartesian Products and Relations
Cartesian Products and Relations Definition (Cartesian product) If A and B are sets, the Cartesian product of A and B is the set A B = {(a, b) :(a A) and (b B)}. The following points are worth special
More informationABSTRACT ALGEBRA: A STUDY GUIDE FOR BEGINNERS
ABSTRACT ALGEBRA: A STUDY GUIDE FOR BEGINNERS John A. Beachy Northern Illinois University 2014 ii J.A.Beachy This is a supplement to Abstract Algebra, Third Edition by John A. Beachy and William D. Blair
More informationMATH PROBLEMS, WITH SOLUTIONS
MATH PROBLEMS, WITH SOLUTIONS OVIDIU MUNTEANU These are free online notes that I wrote to assist students that wish to test their math skills with some problems that go beyond the usual curriculum. These
More informationa 11 x 1 + a 12 x 2 + + a 1n x n = b 1 a 21 x 1 + a 22 x 2 + + a 2n x n = b 2.
Chapter 1 LINEAR EQUATIONS 1.1 Introduction to linear equations A linear equation in n unknowns x 1, x,, x n is an equation of the form a 1 x 1 + a x + + a n x n = b, where a 1, a,..., a n, b are given
More informationT ( a i x i ) = a i T (x i ).
Chapter 2 Defn 1. (p. 65) Let V and W be vector spaces (over F ). We call a function T : V W a linear transformation form V to W if, for all x, y V and c F, we have (a) T (x + y) = T (x) + T (y) and (b)
More informationIntroduction 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
More information26 Ideals and Quotient Rings
Arkansas Tech University MATH 4033: Elementary Modern Algebra Dr. Marcel B. Finan 26 Ideals and Quotient Rings In this section we develop some theory of rings that parallels the theory of groups discussed
More informationProperties of Real Numbers
16 Chapter P Prerequisites P.2 Properties of Real Numbers What you should learn: Identify and use the basic properties of real numbers Develop and use additional properties of real numbers Why you should
More information2. Let H and K be subgroups of a group G. Show that H K G if and only if H K or K H.
Math 307 Abstract Algebra Sample final examination questions with solutions 1. Suppose that H is a proper subgroup of Z under addition and H contains 18, 30 and 40, Determine H. Solution. Since gcd(18,
More informationMatrix Representations of Linear Transformations and Changes of Coordinates
Matrix Representations of Linear Transformations and Changes of Coordinates 01 Subspaces and Bases 011 Definitions A subspace V of R n is a subset of R n that contains the zero element and is closed under
More informationLinear Maps. Isaiah Lankham, Bruno Nachtergaele, Anne Schilling (February 5, 2007)
MAT067 University of California, Davis Winter 2007 Linear Maps Isaiah Lankham, Bruno Nachtergaele, Anne Schilling (February 5, 2007) As we have discussed in the lecture on What is Linear Algebra? one of
More information3. Prime and maximal ideals. 3.1. Definitions and Examples.
COMMUTATIVE ALGEBRA 5 3.1. Definitions and Examples. 3. Prime and maximal ideals Definition. An ideal P in a ring A is called prime if P A and if for every pair x, y of elements in A\P we have xy P. Equivalently,
More informationGROUPS 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
More informationQuotient Rings and Field Extensions
Chapter 5 Quotient Rings and Field Extensions In this chapter we describe a method for producing field extension of a given field. If F is a field, then a field extension is a field K that contains F.
More information1 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
More information4. Matrix inverses. left and right inverse. linear independence. nonsingular matrices. matrices with linearly independent columns
L. Vandenberghe EE133A (Spring 2016) 4. Matrix inverses left and right inverse linear independence nonsingular matrices matrices with linearly independent columns matrices with linearly independent rows
More informationThe Rational Numbers
Math 3040: Spring 2011 The Rational Numbers Contents 1. The Set Q 1 2. Addition and multiplication of rational numbers 3 2.1. Definitions and properties. 3 2.2. Comments 4 2.3. Connections with Z. 6 2.4.
More informationGeometric Transformations
Geometric Transformations Definitions Def: f is a mapping (function) of a set A into a set B if for every element a of A there exists a unique element b of B that is paired with a; this pairing is denoted
More informationFactoring Polynomials
Factoring Polynomials Sue Geller June 19, 2006 Factoring polynomials over the rational numbers, real numbers, and complex numbers has long been a standard topic of high school algebra. With the advent
More informationSection IV.21. The Field of Quotients of an Integral Domain
IV.21 Field of Quotients 1 Section IV.21. The Field of Quotients of an Integral Domain Note. This section is a homage to the rational numbers! Just as we can start with the integers Z and then build the
More informationit 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
More informationLinear 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.
More informationMATRIX ALGEBRA AND SYSTEMS OF EQUATIONS. + + x 2. x n. a 11 a 12 a 1n b 1 a 21 a 22 a 2n b 2 a 31 a 32 a 3n b 3. a m1 a m2 a mn b m
MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS 1. SYSTEMS OF EQUATIONS AND MATRICES 1.1. Representation of a linear system. The general system of m equations in n unknowns can be written a 11 x 1 + a 12 x 2 +
More information4. FIRST STEPS IN THE THEORY 4.1. A
4. FIRST STEPS IN THE THEORY 4.1. A Catalogue of All Groups: The Impossible Dream The fundamental problem of group theory is to systematically explore the landscape and to chart what lies out there. We
More informationMATH 304 Linear Algebra Lecture 9: Subspaces of vector spaces (continued). Span. Spanning set.
MATH 304 Linear Algebra Lecture 9: Subspaces of vector spaces (continued). Span. Spanning set. Vector space A vector space is a set V equipped with two operations, addition V V (x,y) x + y V and scalar
More informationAppendix A. Appendix. A.1 Algebra. Fields and Rings
Appendix A Appendix A.1 Algebra Algebra is the foundation of algebraic geometry; here we collect some of the basic algebra on which we rely. We develop some algebraic background that is needed in the text.
More informationLinear Algebra Notes for Marsden and Tromba Vector Calculus
Linear Algebra Notes for Marsden and Tromba Vector Calculus ndimensional Euclidean Space and Matrices Definition of n space As was learned in Math b, a point in Euclidean three space can be thought of
More informationUniversity 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
More informationCONTENTS 1. Peter Kahn. Spring 2007
CONTENTS 1 MATH 304: CONSTRUCTING THE REAL NUMBERS Peter Kahn Spring 2007 Contents 2 The Integers 1 2.1 The basic construction.......................... 1 2.2 Adding integers..............................
More informationContinued 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
More informationCopy in your notebook: Add an example of each term with the symbols used in algebra 2 if there are any.
Algebra 2  Chapter Prerequisites Vocabulary Copy in your notebook: Add an example of each term with the symbols used in algebra 2 if there are any. P1 p. 1 1. counting(natural) numbers  {1,2,3,4,...}
More informationMATH10040 Chapter 2: Prime and relatively prime numbers
MATH10040 Chapter 2: Prime and relatively prime numbers Recall the basic definition: 1. Prime numbers Definition 1.1. Recall that a positive integer is said to be prime if it has precisely two positive
More information12 Greatest Common Divisors. The Euclidean Algorithm
Arkansas Tech University MATH 4033: Elementary Modern Algebra Dr. Marcel B. Finan 12 Greatest Common Divisors. The Euclidean Algorithm As mentioned at the end of the previous section, we would like to
More informationYou know from calculus that functions play a fundamental role in mathematics.
CHPTER 12 Functions You know from calculus that functions play a fundamental role in mathematics. You likely view a function as a kind of formula that describes a relationship between two (or more) quantities.
More informationThe 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
More informationABSTRACT ALGEBRA. Romyar Sharifi
ABSTRACT ALGEBRA Romyar Sharifi Contents Introduction 7 Part 1. A First Course 11 Chapter 1. Set theory 13 1.1. Sets and functions 13 1.2. Relations 15 1.3. Binary operations 19 Chapter 2. Group theory
More informationINTRODUCTORY SET THEORY
M.Sc. program in mathematics INTRODUCTORY SET THEORY Katalin Károlyi Department of Applied Analysis, Eötvös Loránd University H1088 Budapest, Múzeum krt. 68. CONTENTS 1. SETS Set, equal sets, subset,
More informationFACTORING IN QUADRATIC FIELDS. 1. Introduction. This is called a quadratic field and it has degree 2 over Q. Similarly, set
FACTORING IN QUADRATIC FIELDS KEITH CONRAD For a squarefree integer d other than 1, let 1. Introduction K = Q[ d] = {x + y d : x, y Q}. This is called a quadratic field and it has degree 2 over Q. Similarly,
More informationFinite dimensional C algebras
Finite dimensional C algebras S. Sundar September 14, 2012 Throughout H, K stand for finite dimensional Hilbert spaces. 1 Spectral theorem for selfadjoint opertors Let A B(H) and let {ξ 1, ξ 2,, ξ n
More informationSolutions to TOPICS IN ALGEBRA I.N. HERSTEIN. Part II: Group Theory
Solutions to TOPICS IN ALGEBRA I.N. HERSTEIN Part II: Group Theory No rights reserved. Any part of this work can be reproduced or transmitted in any form or by any means. Version: 1.1 Release: Jan 2013
More information( ) which must be a vector
MATH 37 Linear Transformations from Rn to Rm Dr. Neal, WKU Let T : R n R m be a function which maps vectors from R n to R m. Then T is called a linear transformation if the following two properties are
More informationGroup 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
More information1 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
More information4.6 Null Space, Column Space, Row Space
NULL SPACE, COLUMN SPACE, ROW SPACE Null Space, Column Space, Row Space In applications of linear algebra, subspaces of R n typically arise in one of two situations: ) as the set of solutions of a linear
More informationLINEAR ALGEBRA W W L CHEN
LINEAR ALGEBRA W W L CHEN c W W L Chen, 1997, 2008 This chapter is available free to all individuals, on understanding that it is not to be used for financial gain, and may be downloaded and/or photocopied,
More informationApplications of Fermat s Little Theorem and Congruences
Applications of Fermat s Little Theorem and Congruences Definition: Let m be a positive integer. Then integers a and b are congruent modulo m, denoted by a b mod m, if m (a b). Example: 3 1 mod 2, 6 4
More informationFactoring 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
More information(a) Write each of p and q as a polynomial in x with coefficients in Z[y, z]. deg(p) = 7 deg(q) = 9
Homework #01, due 1/20/10 = 9.1.2, 9.1.4, 9.1.6, 9.1.8, 9.2.3 Additional problems for study: 9.1.1, 9.1.3, 9.1.5, 9.1.13, 9.2.1, 9.2.2, 9.2.4, 9.2.5, 9.2.6, 9.3.2, 9.3.3 9.1.1 (This problem was not assigned
More informationSMALL 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
More information1 Homework 1. [p 0 q i+j +... + p i 1 q j+1 ] + [p i q j ] + [p i+1 q j 1 +... + p i+j q 0 ]
1 Homework 1 (1) Prove the ideal (3,x) is a maximal ideal in Z[x]. SOLUTION: Suppose we expand this ideal by including another generator polynomial, P / (3, x). Write P = n + x Q with n an integer not
More informationDiscrete Mathematics: Solutions to Homework (12%) For each of the following sets, determine whether {2} is an element of that set.
Discrete Mathematics: Solutions to Homework 2 1. (12%) For each of the following sets, determine whether {2} is an element of that set. (a) {x R x is an integer greater than 1} (b) {x R x is the square
More information= 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
More information3 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
More informationNOTES ON LINEAR TRANSFORMATIONS
NOTES ON LINEAR TRANSFORMATIONS Definition 1. Let V and W be vector spaces. A function T : V W is a linear transformation from V to W if the following two properties hold. i T v + v = T v + T v for all
More information1 = (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 realclosed fields These notes develop the algebraic background needed to understand the model theory of realclosed fields. To understand these notes, a standard graduate course in algebra is
More informationMath 223 Abstract Algebra Lecture Notes
Math 223 Abstract Algebra Lecture Notes Steven Tschantz Spring 2001 (Apr. 23 version) Preamble These notes are intended to supplement the lectures and make up for the lack of a textbook for the course
More informationGROUP 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
More informationGroup Fundamentals. Chapter 1. 1.1 Groups and Subgroups. 1.1.1 Definition
Chapter 1 Group Fundamentals 1.1 Groups and Subgroups 1.1.1 Definition A group is a nonempty set G on which there is defined a binary operation (a, b) ab satisfying the following properties. Closure: If
More informationMath 319 Problem Set #3 Solution 21 February 2002
Math 319 Problem Set #3 Solution 21 February 2002 1. ( 2.1, problem 15) Find integers a 1, a 2, a 3, a 4, a 5 such that every integer x satisfies at least one of the congruences x a 1 (mod 2), x a 2 (mod
More informationProblem Set 7  Fall 2008 Due Tuesday, Oct. 28 at 1:00
18.781 Problem Set 7  Fall 2008 Due Tuesday, Oct. 28 at 1:00 Throughout this assignment, f(x) always denotes a polynomial with integer coefficients. 1. (a) Show that e 32 (3) = 8, and write down a list
More informationKevin James. MTHSC 412 Section 2.4 Prime Factors and Greatest Comm
MTHSC 412 Section 2.4 Prime Factors and Greatest Common Divisor Greatest Common Divisor Definition Suppose that a, b Z. Then we say that d Z is a greatest common divisor (gcd) of a and b if the following
More informationMath 313 Lecture #10 2.2: The Inverse of a Matrix
Math 1 Lecture #10 2.2: The Inverse of a Matrix Matrix algebra provides tools for creating many useful formulas just like real number algebra does. For example, a real number a is invertible if there is
More informationMATRIX ALGEBRA AND SYSTEMS OF EQUATIONS
MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS Systems of Equations and Matrices Representation of a linear system The general system of m equations in n unknowns can be written a x + a 2 x 2 + + a n x n b a
More informationsome 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
More informationZORN 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 nonconstructive existence theorems throughout mathematics. We will
More informationMODULAR ARITHMETIC. a smallest member. It is equivalent to the Principle of Mathematical Induction.
MODULAR ARITHMETIC 1 Working With Integers The usual arithmetic operations of addition, subtraction and multiplication can be performed on integers, and the result is always another integer Division, on
More informationGroup Theory (MA343): Lecture Notes Semester I Dr Rachel Quinlan School of Mathematics, Statistics and Applied Mathematics, NUI Galway
Group Theory (MA343): Lecture Notes Semester I 20132014 Dr Rachel Quinlan School of Mathematics, Statistics and Applied Mathematics, NUI Galway November 21, 2013 Contents 1 What is a group? 2 1.1 Examples...........................................
More information5. Linear algebra I: dimension
5. Linear algebra I: dimension 5.1 Some simple results 5.2 Bases and dimension 5.3 Homomorphisms and dimension 1. Some simple results Several observations should be made. Once stated explicitly, the proofs
More informationLinear Algebra. A vector space (over R) is an ordered quadruple. such that V is a set; 0 V ; and the following eight axioms hold:
Linear Algebra A vector space (over R) is an ordered quadruple (V, 0, α, µ) such that V is a set; 0 V ; and the following eight axioms hold: α : V V V and µ : R V V ; (i) α(α(u, v), w) = α(u, α(v, w)),
More informationHomework 5 Solutions
Homework 5 Solutions 4.2: 2: a. 321 = 256 + 64 + 1 = (01000001) 2 b. 1023 = 512 + 256 + 128 + 64 + 32 + 16 + 8 + 4 + 2 + 1 = (1111111111) 2. Note that this is 1 less than the next power of 2, 1024, which
More information1. The algebra of exponents 1.1. Natural Number Powers. It is easy to say what is meant by a n a (raised to) to the (power) n if n N.
CHAPTER 3: EXPONENTS AND POWER FUNCTIONS 1. The algebra of exponents 1.1. Natural Number Powers. It is easy to say what is meant by a n a (raised to) to the (power) n if n N. For example: In general, if
More informationPolynomial Invariants
Polynomial Invariants Dylan Wilson October 9, 2014 (1) Today we will be interested in the following Question 1.1. What are all the possible polynomials in two variables f(x, y) such that f(x, y) = f(y,
More informationB such that AB = I and BA = I. (We say B is an inverse of A.) Definition A square matrix A is invertible (or nonsingular) if matrix
Matrix inverses Recall... Definition A square matrix A is invertible (or nonsingular) if matrix B such that AB = and BA =. (We say B is an inverse of A.) Remark Not all square matrices are invertible.
More informationSection 6.1  Inner Products and Norms
Section 6.1  Inner Products and Norms Definition. Let V be a vector space over F {R, C}. An inner product on V is a function that assigns, to every ordered pair of vectors x and y in V, a scalar in F,
More informationMath 4310 Handout  Quotient Vector Spaces
Math 4310 Handout  Quotient Vector Spaces Dan Collins The textbook defines a subspace of a vector space in Chapter 4, but it avoids ever discussing the notion of a quotient space. This is understandable
More information3. Applications of Number Theory
3. APPLICATIONS OF NUMBER THEORY 163 3. Applications of Number Theory 3.1. Representation of Integers. Theorem 3.1.1. Given an integer b > 1, every positive integer n can be expresses uniquely as n = a
More informationThe determinant of a skewsymmetric 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
More informationBasic Concepts of Point Set Topology Notes for OU course Math 4853 Spring 2011
Basic Concepts of Point Set Topology Notes for OU course Math 4853 Spring 2011 A. Miller 1. Introduction. The definitions of metric space and topological space were developed in the early 1900 s, largely
More informationMatrix Algebra. Some Basic Matrix Laws. Before reading the text or the following notes glance at the following list of basic matrix algebra laws.
Matrix Algebra A. Doerr Before reading the text or the following notes glance at the following list of basic matrix algebra laws. Some Basic Matrix Laws Assume the orders of the matrices are such that
More informationPUTNAM 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
More information2.1: MATRIX OPERATIONS
.: MATRIX OPERATIONS What are diagonal entries and the main diagonal of a matrix? What is a diagonal matrix? When are matrices equal? Scalar Multiplication 45 Matrix Addition Theorem (pg 0) Let A, B, and
More information4: EIGENVALUES, EIGENVECTORS, DIAGONALIZATION
4: EIGENVALUES, EIGENVECTORS, DIAGONALIZATION STEVEN HEILMAN Contents 1. Review 1 2. Diagonal Matrices 1 3. Eigenvectors and Eigenvalues 2 4. Characteristic Polynomial 4 5. Diagonalizability 6 6. Appendix:
More informationby 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 yaxis We observe that
More informationTHE 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
More informationRow 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
More information