# Rings. EE 387, Notes 7, Handout #10

Size: px
Start display at page:

Transcription

1 Rings EE 387, Notes 7, Handout #10 Definition: A ring is a set R with binary operations, + and, that satisfy the following axioms: 1. (R, +) is a commutative group (five axioms) 2. Associative law for multiplication: (a b) c = a (b c) 3. Distributive laws: a (b+c) = (a b)+(a c), (b+c) a = (b a)+(c a) (Two distributive laws are needed if multiplication is not commutative.) Here is an example of an obvious property that holds for all rings. Proposition: In any ring, 0 a = 0. Proof: By the distributive law, 0 a = (0+0) a = (0 a)+(0 a) Subtracting 0 a from both sides of equation yields 0 = 0 a. EE 387, October 5, 2015 Notes 7, Page 1

2 Important rings Several rings will be used in this course: integers Z = {..., 3, 2, 1,0,+1,+2,+3,...} integers modulo m: Z m = {0,1,...,m 1} polynomials with coefficients from a field: F[x] = {f 0 +f 1 x+ +f n x n : n 0, f i F } polynomials over a field modulo a prime polynomial p(x) of degree m the n n matrices with coefficients from a field Similarities and differences between the rings of integers and of binary polynomials. Similarities: Elements can be represented by bit strings Multiplication by shift-and-add algorithms Differences: Arithmetic for polynomials does not require carries Factoring binary polynomials is easy but factoring integers seems hard. EE 387, October 5, 2015 Notes 7, Page 2

3 Rings with additional properties By adding more requirements to rings, we ultimately arrive at fields. Commutative ring: a b = b a. The 2 2 matrices are a familiar example of a noncommutative ring: [ ][ ] [ ] [ ] [ ][ ] = = Some subrings of 2 2 matrices are commutative. E.g., complex numbers: [ ][ ] [ ] [ ][ ] a b c d ac bd ad bc c d a b = = b a d c ad+bc ac bd d c b a Ring with identity: there is an element 1 such that 1 a = a 1 = a. Ring without identity: even integers 2Z = {..., 4, 2,0,+2,+4,...}. Fact: if an identity exists, the identity is unique. Proof: 1 1 = = 1 2. (Same as proof for groups.) EE 387, October 5, 2015 Notes 7, Page 3

4 Inverses and divisors Inverse or reciprocal of a: an element a 1 with a a 1 = a 1 a = 1. If it exists, an inverse is unique. An element with an inverse is called a unit. Units in familiar rings: integers: ±1. polynomials over a field: nonzero constants. matrices over a field: matrices with nonzero determinant matrices with integer coefficients: units are? Divisor: if c = a b then a and b are divisors of c. The units in a ring are the divisors of 1. A nonunit c is irreducible if whenever c = a b then either a or b is a unit. Zero divisor: if a b = 0 but a 0 and b 0 then a and b are zero divisors. The integers and the polynomials over a field have no zero divisors. A zero divisor does not have an inverse and is therefore not a unit. EE 387, October 5, 2015 Notes 7, Page 4

5 Integral domain, division ring, field Integral domain: a commutative ring without zero divisors. Integral domains have many properties in common with the integers. The integers are a subring of the field of rational numbers: { p } Q = q : p,q Z, q 0 Similarly, the polynomials are subring of the field of rational functions: { p(x) } F(x) = q(x) : p(x),q(x) F[x], q(x) 0 Fact: quotients of elements from any integral domain form a field. Division ring or skew field: noncommutative ring in which every nonzero element has a multiplicative inverse. Most famous skew field: Sir William Hamilton s quaternions. These extend the complex numbers obtained by adding imaginary elements, j and k, such that i 2 = j 2 = k 2 = ijk = 1. Field: a commutative division ring. EE 387, October 5, 2015 Notes 7, Page 5

6 Fields, polynomials, and vector spaces A field is a set F with associative and commutative operators + and that satisfy the following axioms. 1. (F, +) is a commutative group 2. (F {0}, ) is a commutative group 3. Distributive law: a (b+c) = (a b)+(a c) = ab+ac Fields have the 4 basic arithmetic operations: +,, and inverses,. Infinite fields: rational numbers Q, algebraic numbers, real numbers R, complex numbers C, rational functions. Finite fields (all possible): GF(p) = integers with arithmetic modulo a prime number p GF(p m ) = polynomials over GF(p) with arithmetic modulo a prime polynomial of degree m An algebraic number is any zero of polynomial with integer coefficients; e.g., 3 3 or (1+ 5)/2. EE 387, October 5, 2015 Notes 7, Page 6

7 The finite field GF(p) If p is prime, there is one and only one field with p elements: Z p = {0, 1, 1+1,..., }{{} } = {0, 1,..., p 1} p 1 with addition and multiplication mod p. This field is named GF(p). Associativity and commutativity of addition and multiplication mod p follow from the corresponding properties for integer arithmetic. The only field axiom that remains to be verified is the existence of multiplicative inverses. Recall that if p is a prime number and 0 < r < p, then gcd(r,p) = 1. Since gcd(r,p) = ar +bp for some integers a and b, 1 = gcd(r,p) = ar+bp = ar = 1 bp 1 mod p Thus the reciprocal of r in GF(p) is a mod p. EE 387, October 5, 2015 Notes 7, Page 7

8 Polynomials The polynomials over a field F are F[x], the expressions of the form f 0 +f 1 x+f 2 x 2 + +f n x n, n N, f i F F[x] is a commutative ring with identity and no zero divisors, i.e., an integral domain. Different views of polynomial f(x) with coefficients from a field F: Function: x f 0 +f 1 x+ +f n x n where x,f 0,...,f n belong to F Finite power series: f 0 +f 1 x+ +f n 1 x n 1 +f n x n +0x n+1 + Finite sequence: (f 0,f 1,...,f n 1,f n ) with coefficients from F Infinite sequence, finitely many nonzero terms: (f 0,f 1,...,f n,0,0,...) The functional view is not adequate for finite fields; there are infinitely many binary polynomials but only 2 2 = 4 functions from {0,1} to {0,1}. Example: nonzero polynomial x 2 +x over GF(2) evaluates to 0 for x = 0,1, which are the only two elements of GF(2). EE 387, October 5, 2015 Notes 7, Page 8

9 Facts about polynomials Definition: The degree of a polynomial is the exponent of the term with the highest power of the indeterminant. Well known facts about degrees of polynomials: deg(f(x)±g(x)) max(degf(x),degg(x)) degf(x)g(x) = degf(x)+degg(x), since (f n x n + +f 0 )(g m x m + +g 0 ) = f n g m x n+m + +f 0 g 0 degf(x)/g(x) = degf(x) degg(x) if g(x) 0 Because of the convention deg0 =, degf(x)g(x) = degf(x)+degg(x) is true even when one of the factors is 0. By convention, the degree of 0 is. EE 387, October 5, 2015 Notes 7, Page 9

10 Prime polynomials Definition: A monic polynomial is a polynomial with leading coefficient 1. Example: x 3 +2x 2 +3x+4 is monic, 2x 3 +4x 2 +6x+8 is not. Every monic polynomial is nonzero. All nonzero polynomials over GF(2) are monic. Definition: An irreducible polynomial is a polynomial that has no proper divisors (divisors of smaller degree). Definition: A prime polynomial is a monic irreducible polynomial. Example: Over the real numbers, 2x 2 +1 is irreducible, but x is prime. 2 Over the complex numbers, x ( 2 = x+ i )( x i ) is not prime. 2 2 Note: irreducibility depends on what coefficients are allowed in the factors. EE 387, October 5, 2015 Notes 7, Page 10

11 Divison algorithm for polynomials For every dividend a(x) F[x] and nonzero divisor d(x) F[x] there is a unique quotient q(x) and remainder r(x) such that a(x) = q(x)d(x)+r(x), where degr(x) < degd(x) Multiples and divisors are defined for polynomials just as for integers. The greatest common divisor gcd(r(x), s(x)) is the monic polynomial of largest degree that is a common divisor of smallest degree of the form a(x)r(x)+b(x)s(x) If p(x) is prime and 0 degr(x) < degp(x), then gcd(r(x),p(x)) = 1. a(x)r(x)+b(x)p(x) = 1 = a(x)r(x) mod p(x) = 1 Thus r(x) has a reciprocal. = a(x) = r(x) 1 mod p(x). Polynomials over F of degree < m with arithmetic mod p(x) form a field. If F is GF(p), this finite field GF(p m ) consists of m-tuples from GF(p). EE 387, October 5, 2015 Notes 7, Page 11

12 Finite field GF(2 4 ) Over GF(2), is x 4 +x+1 is prime of degree 4. Elements of GF(2 4 ) can be Multiplication table for GF(2 4 ) represented by polynomial expressions or by bit strings (msb first); e.g., 0101 = 1+x = 1+x+x = x 3 Higher powers of x can be reduced using remainder operation: x 4 = x 4 mod (x 4 +x+1) = x+1 x x 3 = x = 0011 This one equation completely determines multiplication in GF(2 4 ) multiplication A B C D E F A B C D E F A C E B 9 F D C F A 9 B 8 D E C 3 7 B F 6 2 E A 5 1 D A F 7 2 D 8 E B C C A B D F E E 9 F D A C B B 6 E 5 D C 4 F 7 A B 3 A 4 D 5 C 6 F 7 E A 0 A 7 D E F B 6 C B 0 B 5 E A 1 F 4 7 C 2 9 D C 0 C B E 2 A 6 1 D F D 0 D C F B 6 3 E A 7 E 0 E F 1 D 3 2 C A B 5 F 0 F D B 1 E C A EE 387, October 5, 2015 Notes 7, Page 12

13 Vector spaces A vector space V of vectors over a field F of scalars is a set with a binary operator + on V and a scalar-vector product satisfying these axioms: 1. (V, +) is a commutative group 2. Associative law: (a 1 a 2 ) v = a 1 (a 2 v). 3. Distributive laws: a (v 1 +v 2 ) = a v 1 +a v 2, (a 1 +a 2 ) v = a 1 v +a 2 v 4. Unitary law: 1 v = v for every vector v in V Recall definition of field: a set F with associative and commutative operators + and that satisfy the following axioms. 1. (F, +) is a commutative group 2. (F {0}, ) is a commutative group 3. Distributive law: a (b+c) = (a b)+(a c) A field is a vector space over itself of dimension 1. In general, vector spaces do not have vector-vector multiplication. An algebra is a vector space with an associative, distributive multiplication; e.g., n n matrices over F. EE 387, October 5, 2015 Notes 7, Page 13

14 Span, basis, linear independence Definition: The span of a set of vectors {v 1,...,v n } is the set of all linear combinations of those vectors: {a 1 v 1 + +a n v n : a 1,...,a n F }. Definition: A set of vectors {v 1,...,v n } is linearly independent (LI) if no vector in the set is a linear combination of the other vectors; equivalently, there does not exist a set of scalars {a 1,...,a n }, not all zero, such that a 1 v 1 + a 2 v a n v n = 0. Definition: A basis for a vector space is a LI set that spans the vector space. Definition: A vector space is finite dimensional if it has a finite basis. Lemma: The number of vectors in a linearly independent set is the number of vectors in a spanning set. Theorem: All bases for a finite-dimensional vector space have the same number of elements. Thus the dimension of the vector space is well defined. EE 387, October 5, 2015 Notes 7, Page 14

15 Vector coordinates Theorem: Every vector can be expressed as a linear combination of basis vectors in exactly one way. Proof: Suppose that {v 1,...,v n } is a basis and a 1 v a n v n = a 1 v a n v n, are two possibly different representations of the same vector. Then (a 1 a 1) v 1 + +(a n a n) v n = 0. By linear independence of the basis a 1 a 1 = 0,...,a n a n = 0 = a 1 = a 1,...,a n = a n. We say that V is isomorphic to vector space F n = {(a 1,...,a n ) : a i F} with componentwise addition and scalar multiplication. Theorem: If V is a vector space of dimension n over a field with q elements, then the number of elements of V is q n. Proof: There are q n n-tuples of scalars from F. EE 387, October 5, 2015 Notes 7, Page 15

16 Inner product and orthogonal complement Definition: A vector subspace of a vector space V is a subset W that is a vector space, i.e., closed under vector addition and scalar multiplication. Theorem: If W is a subspace of V then dimw dimv. Proof: Any basis {w 1,...,w k } for subspace W can be extended to a basis of V by adding n k vectors {v k+1,...,v n }. Therefore dimw dimv. Definition: The inner product of two n-tuples over a field F is (a 1,...,a n ) (b 1,...,b n ) = a 1 b 1 + +a n b n. Definition: Two n-tuples are orthogonal if their inner product is zero. Definition: The orthogonal complement of a subspace W of n-tuples is the set W of vectors orthogonal to every vector in W; that is, v W if and only if v w = 0 for every w W. Theorem: (Dimension Theorem) If dimw = k then dimw = n k. Theorem: If W is subspace of finite-dimensional V, then W = W. In infinite-dimensional vector spaces, W may be a proper subset of W. EE 387, October 5, 2015 Notes 7, Page 16

17 Linear transformations and matrix multiplication Definition: A linear transformation on a vector space V is a function T that maps vectors in V to vectors in V such that for all vectors v 1,v 2 and scalars a 1,a 2, T(a 1 v 1 +a 2 v 2 ) = a 1 T(v 1 )+a 2 T(v 2 ). Every linear transformation T can be described by an n n matrix M. The coordinates of T(v 1,...,v n ) are the components of the vector-matrix product (v 1,...,v n )M. Each component of w = vm is inner product of v with a column of M: w j = v 1 m 1j + +v n m nj = n i=1 v im ij. Here is another way to look at w = vm. If m i is i-th row of M, then (v 1,...,v n )M = v 1 m 1 + +v n m n. Thus vm is linear combination of rows of M weighted by coordinates of v. As v ranges over n-tuples, vm ranges over subspace spanned by rows of M. EE 387, October 5, 2015 Notes 7, Page 17

18 Matrix rank Definition: A square matrix is nonsingular if it has a multiplicative inverse. Definition: The determinant of A = (a ij ) is deta = ( 1) σ a i1 σ(i 1 ) a inσ(i n), σ where σ is a permutation of {1,...,n} and ( 1) σ is the sign of σ. Theorem: A matrix is nonsingular if and only if its determinant is nonzero. Definition: The rank of a matrix is the size of largest nonsingular submatrix. The rank of a matrix can be computed efficiently: 1. Use Gaussian elimination to transform matrix to row-reduced echelon form. 2. The rank is the number of nonzero rows of the row-reduced echelon matrix. Theorem: The rank of a matrix is the dimension of the row space, which equals the dimension of the column space. Therefore the number of linearly independent rows is the same as the number of linearly independent columns. EE 387, October 5, 2015 Notes 7, Page 18

### Vector spaces. EE 387, Notes 8, Handout #12

Vector spaces EE 387, Notes 8, Handout #12 A vector space V of vectors over a field F of scalars is a set with a binary operator + on V and a scalar-vector product satisfying these axioms: 1. (V, +) is

### Polynomials. Chapter 4

Chapter 4 Polynomials In this Chapter we shall see that everything we did with integers in the last Chapter we can also do with polynomials. Fix a field F (e.g. F = Q, R, C or Z/(p) for a prime p). Notation

### MTH310 EXAM 2 REVIEW

MTH310 EXAM 2 REVIEW SA LI 4.1 Polynomial Arithmetic and the Division Algorithm A. Polynomial Arithmetic *Polynomial Rings If R is a ring, then there exists a ring T containing an element x that is not

### CHAPTER I. Rings. Definition A ring R is a set with two binary operations, addition + and

CHAPTER I Rings 1.1 Definitions and Examples Definition 1.1.1. A ring R is a set with two binary operations, addition + and multiplication satisfying the following conditions for all a, b, c in R : (i)

### Algebra Review 2. 1 Fields. A field is an extension of the concept of a group.

Algebra Review 2 1 Fields A field is an extension of the concept of a group. Definition 1. A field (F, +,, 0 F, 1 F ) is a set F together with two binary operations (+, ) on F such that the following conditions

### A field F is a set of numbers that includes the two numbers 0 and 1 and satisfies the properties:

Byte multiplication 1 Field arithmetic A field F is a set of numbers that includes the two numbers 0 and 1 and satisfies the properties: F is an abelian group under addition, meaning - F is closed under

### + 1 3 x2 2x x3 + 3x 2 + 0x x x2 2x + 3 4

Math 4030-001/Foundations of Algebra/Fall 2017 Polynomials at the Foundations: Rational Coefficients The rational numbers are our first field, meaning that all the laws of arithmetic hold, every number

### Fundamental Theorem of Algebra

EE 387, Notes 13, Handout #20 Fundamental Theorem of Algebra Lemma: If f(x) is a polynomial over GF(q) GF(Q), then β is a zero of f(x) if and only if x β is a divisor of f(x). Proof: By the division algorithm,

### Rings 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

### Commutative 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

### 0.2 Vector spaces. J.A.Beachy 1

J.A.Beachy 1 0.2 Vector spaces I m going to begin this section at a rather basic level, giving the definitions of a field and of a vector space in much that same detail as you would have met them in a

### Linear Algebra. Workbook

Linear Algebra Workbook Paul Yiu Department of Mathematics Florida Atlantic University Last Update: November 21 Student: Fall 2011 Checklist Name: A B C D E F F G H I J 1 2 3 4 5 6 7 8 9 10 xxx xxx xxx

### RINGS: 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

### GRE Subject test preparation Spring 2016 Topic: Abstract Algebra, Linear Algebra, Number Theory.

GRE Subject test preparation Spring 2016 Topic: Abstract Algebra, Linear Algebra, Number Theory. Linear Algebra Standard matrix manipulation to compute the kernel, intersection of subspaces, column spaces,

### Polynomial Rings. i=0

Polynomial Rings 4-15-2018 If R is a ring, the ring of polynomials in x with coefficients in R is denoted R[x]. It consists of all formal sums a i x i. Here a i = 0 for all but finitely many values of

### 2 so Q[ 2] is closed under both additive and multiplicative inverses. a 2 2b 2 + b

. FINITE-DIMENSIONAL VECTOR SPACES.. Fields By now you ll have acquired a fair knowledge of matrices. These are a concrete embodiment of something rather more abstract. Sometimes it is easier to use matrices,

### Algebra Homework, Edition 2 9 September 2010

Algebra Homework, Edition 2 9 September 2010 Problem 6. (1) Let I and J be ideals of a commutative ring R with I + J = R. Prove that IJ = I J. (2) Let I, J, and K be ideals of a principal ideal domain.

### Section III.6. Factorization in Polynomial Rings

III.6. Factorization in Polynomial Rings 1 Section III.6. Factorization in Polynomial Rings Note. We push several of the results in Section III.3 (such as divisibility, irreducibility, and unique factorization)

### Chapter 4 Finite Fields

Chapter 4 Finite Fields Introduction will now introduce finite fields of increasing importance in cryptography AES, Elliptic Curve, IDEA, Public Key concern operations on numbers what constitutes a number

### Outline. MSRI-UP 2009 Coding Theory Seminar, Week 2. The definition. Link to polynomials

Outline MSRI-UP 2009 Coding Theory Seminar, Week 2 John B. Little Department of Mathematics and Computer Science College of the Holy Cross Cyclic Codes Polynomial Algebra More on cyclic codes Finite fields

### 7.1 Definitions and Generator Polynomials

Chapter 7 Cyclic Codes Lecture 21, March 29, 2011 7.1 Definitions and Generator Polynomials Cyclic codes are an important class of linear codes for which the encoding and decoding can be efficiently implemented

### COMPUTER ARITHMETIC. 13/05/2010 cryptography - math background pp. 1 / 162

COMPUTER ARITHMETIC 13/05/2010 cryptography - math background pp. 1 / 162 RECALL OF COMPUTER ARITHMETIC computers implement some types of arithmetic for instance, addition, subtratction, multiplication

### Abstract Algebra: Chapters 16 and 17

Study polynomials, their factorization, and the construction of fields. Chapter 16 Polynomial Rings Notation Let R be a commutative ring. The ring of polynomials over R in the indeterminate x is the set

### Kevin James. MTHSC 412 Section 3.1 Definition and Examples of Rings

MTHSC 412 Section 3.1 Definition and Examples of Rings A ring R is a nonempty set R together with two binary operations (usually written as addition and multiplication) that satisfy the following axioms.

### Introduction to finite fields

Chapter 7 Introduction to finite fields This chapter provides an introduction to several kinds of abstract algebraic structures, particularly groups, fields, and polynomials. Our primary interest is in

### 1. Group Theory Permutations.

1.1. Permutations. 1. Group Theory Problem 1.1. Let G be a subgroup of S n of index 2. Show that G = A n. Problem 1.2. Find two elements of S 7 that have the same order but are not conjugate. Let π S 7

### Course 311: Michaelmas Term 2005 Part III: Topics in Commutative Algebra

Course 311: Michaelmas Term 2005 Part III: Topics in Commutative Algebra D. R. Wilkins Contents 3 Topics in Commutative Algebra 2 3.1 Rings and Fields......................... 2 3.2 Ideals...............................

### CHAPTER 14. Ideals and Factor Rings

CHAPTER 14 Ideals and Factor Rings Ideals Definition (Ideal). A subring A of a ring R is called a (two-sided) ideal of R if for every r 2 R and every a 2 A, ra 2 A and ar 2 A. Note. (1) A absorbs elements

### Lecture 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

### Prime 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

### Mathematical Foundations of Cryptography

Mathematical Foundations of Cryptography Cryptography is based on mathematics In this chapter we study finite fields, the basis of the Advanced Encryption Standard (AES) and elliptical curve cryptography

### 6x 3 12x 2 7x 2 +16x 7x 2 +14x 2x 4

2.3 Real Zeros of Polynomial Functions Name: Pre-calculus. Date: Block: 1. Long Division of Polynomials. We have factored polynomials of degree 2 and some specific types of polynomials of degree 3 using

### Dividing Polynomials: Remainder and Factor Theorems

Dividing Polynomials: Remainder and Factor Theorems When we divide one polynomial by another, we obtain a quotient and a remainder. If the remainder is zero, then the divisor is a factor of the dividend.

### Rings. Chapter 1. Definition 1.2. A commutative ring R is a ring in which multiplication is commutative. That is, ab = ba for all a, b R.

Chapter 1 Rings We have spent the term studying groups. A group is a set with a binary operation that satisfies certain properties. But many algebraic structures such as R, Z, and Z n come with two binary

### Polynomial Rings. (Last Updated: December 8, 2017)

Polynomial Rings (Last Updated: December 8, 2017) These notes are derived primarily from Abstract Algebra, Theory and Applications by Thomas Judson (16ed). Most of this material is drawn from Chapters

### Polynomial Rings. i=0. i=0. n+m. i=0. k=0

Polynomial Rings 1. Definitions and Basic Properties For convenience, the ring will always be a commutative ring with identity. Basic Properties The polynomial ring R[x] in the indeterminate x with coefficients

Mathematical Olympiad Training Polynomials Definition A polynomial over a ring R(Z, Q, R, C) in x is an expression of the form p(x) = a n x n + a n 1 x n 1 + + a 1 x + a 0, a i R, for 0 i n. If a n 0,

### SYMBOL EXPLANATION EXAMPLE

MATH 4310 PRELIM I REVIEW Notation These are the symbols we have used in class, leading up to Prelim I, and which I will use on the exam SYMBOL EXPLANATION EXAMPLE {a, b, c, } The is the way to write the

### Moreover this binary operation satisfies the following properties

Contents 1 Algebraic structures 1 1.1 Group........................................... 1 1.1.1 Definitions and examples............................. 1 1.1.2 Subgroup.....................................

### Rings. Chapter Definitions and Examples

Chapter 5 Rings Nothing proves more clearly that the mind seeks truth, and nothing reflects more glory upon it, than the delight it takes, sometimes in spite of itself, in the driest and thorniest researches

### 2a 2 4ac), provided there is an element r in our

MTH 310002 Test II Review Spring 2012 Absractions versus examples The purpose of abstraction is to reduce ideas to their essentials, uncluttered by the details of a specific situation Our lectures built

### Algebraic 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

### Chapter 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

### Handout - 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

### MATH 3030, Abstract Algebra FALL 2012 Toby Kenney Midyear Examination Friday 7th December: 7:00-10:00 PM

MATH 3030, Abstract Algebra FALL 2012 Toby Kenney Midyear Examination Friday 7th December: 7:00-10:00 PM Basic Questions 1. Compute the factor group Z 3 Z 9 / (1, 6). The subgroup generated by (1, 6) is

### Algebra Exam Syllabus

Algebra Exam Syllabus The Algebra comprehensive exam covers four broad areas of algebra: (1) Groups; (2) Rings; (3) Modules; and (4) Linear Algebra. These topics are all covered in the first semester graduate

### ALGEBRA QUALIFYING EXAM PROBLEMS LINEAR ALGEBRA

ALGEBRA QUALIFYING EXAM PROBLEMS LINEAR ALGEBRA Kent State University Department of Mathematical Sciences Compiled and Maintained by Donald L. White Version: August 29, 2017 CONTENTS LINEAR ALGEBRA AND

### Many of the groups with which we are familiar are arithmetical in nature, and they tend to share key structures that combine more than one operation.

12. Rings 1 Rings Many of the groups with which we are familiar are arithmetical in nature, and they tend to share key structures that combine more than one operation. Example: Z, Q, R, and C are an Abelian

### Coding 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

### Mathematics for Cryptography

Mathematics for Cryptography Douglas R. Stinson David R. Cheriton School of Computer Science University of Waterloo Waterloo, Ontario, N2L 3G1, Canada March 15, 2016 1 Groups and Modular Arithmetic 1.1

### 3. Coding theory 3.1. Basic concepts

3. CODING THEORY 1 3. Coding theory 3.1. Basic concepts In this chapter we will discuss briefly some aspects of error correcting codes. The main problem is that if information is sent via a noisy channel,

### Lecture Summaries for Linear Algebra M51A

These lecture summaries may also be viewed online by clicking the L icon at the top right of any lecture screen. Lecture Summaries for Linear Algebra M51A refers to the section in the textbook. Lecture

### Linear Algebra Practice Problems

Linear Algebra Practice Problems Math 24 Calculus III Summer 25, Session II. Determine whether the given set is a vector space. If not, give at least one axiom that is not satisfied. Unless otherwise stated,

### Finite Fields. Mike Reiter

1 Finite Fields Mike Reiter reiter@cs.unc.edu Based on Chapter 4 of: W. Stallings. Cryptography and Network Security, Principles and Practices. 3 rd Edition, 2003. Groups 2 A group G, is a set G of elements

### Some notes on linear algebra

Some notes on linear algebra Throughout these notes, k denotes a field (often called the scalars in this context). Recall that this means that there are two binary operations on k, denoted + and, that

### Algebra Review. Instructor: Laszlo Babai Notes by Vincent Lucarelli and the instructor. June 15, 2001

Algebra Review Instructor: Laszlo Babai Notes by Vincent Lucarelli and the instructor June 15, 2001 1 Groups Definition 1.1 A semigroup (G, ) is a set G with a binary operation such that: Axiom 1 ( a,

### MATH 431 PART 2: POLYNOMIAL RINGS AND FACTORIZATION

MATH 431 PART 2: POLYNOMIAL RINGS AND FACTORIZATION 1. Polynomial rings (review) Definition 1. A polynomial f(x) with coefficients in a ring R is n f(x) = a i x i = a 0 + a 1 x + a 2 x 2 + + a n x n i=0

### Groups, 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

### Cyclic codes: overview

Cyclic codes: overview EE 387, Notes 14, Handout #22 A linear block code is cyclic if the cyclic shift of a codeword is a codeword. Cyclic codes have many advantages. Elegant algebraic descriptions: c(x)

### x n k m(x) ) Codewords can be characterized by (and errors detected by): c(x) mod g(x) = 0 c(x)h(x) = 0 mod (x n 1)

Cyclic codes: review EE 387, Notes 15, Handout #26 A cyclic code is a LBC such that every cyclic shift of a codeword is a codeword. A cyclic code has generator polynomial g(x) that is a divisor of every

### Finite Fields. Saravanan Vijayakumaran Department of Electrical Engineering Indian Institute of Technology Bombay

1 / 25 Finite Fields Saravanan Vijayakumaran sarva@ee.iitb.ac.in Department of Electrical Engineering Indian Institute of Technology Bombay September 25, 2014 2 / 25 Fields Definition A set F together

### AN INTRODUCTION TO THE THEORY OF FIELD EXTENSIONS

AN INTRODUCTION TO THE THEORY OF FIELD EXTENSIONS SAMUEL MOY Abstract. Assuming some basic knowledge of groups, rings, and fields, the following investigation will introduce the reader to the theory of

### G Solution (10 points) Using elementary row operations, we transform the original generator matrix as follows.

EE 387 October 28, 2015 Algebraic Error-Control Codes Homework #4 Solutions Handout #24 1. LBC over GF(5). Let G be a nonsystematic generator matrix for a linear block code over GF(5). 2 4 2 2 4 4 G =

Algebra Qualifying Exam August 2001 Do all 5 problems. 1. Let G be afinite group of order 504 = 23 32 7. a. Show that G cannot be isomorphic to a subgroup of the alternating group Alt 7. (5 points) b.

### School of Mathematics and Statistics. MT5836 Galois Theory. Handout 0: Course Information

MRQ 2017 School of Mathematics and Statistics MT5836 Galois Theory Handout 0: Course Information Lecturer: Martyn Quick, Room 326. Prerequisite: MT3505 (or MT4517) Rings & Fields Lectures: Tutorials: Mon

### Abstract Algebra, Second Edition, by John A. Beachy and William D. Blair. Corrections and clarifications

1 Abstract Algebra, Second Edition, by John A. Beachy and William D. Blair Corrections and clarifications Note: Some corrections were made after the first printing of the text. page 9, line 8 For of the

### CSL361 Problem set 4: Basic linear algebra

CSL361 Problem set 4: Basic linear algebra February 21, 2017 [Note:] If the numerical matrix computations turn out to be tedious, you may use the function rref in Matlab. 1 Row-reduced echelon matrices

### Part IX. Factorization

IX.45. Unique Factorization Domains 1 Part IX. Factorization Section IX.45. Unique Factorization Domains Note. In this section we return to integral domains and concern ourselves with factoring (with respect

### IRREDUCIBLES AND PRIMES IN COMPUTABLE INTEGRAL DOMAINS

IRREDUCIBLES AND PRIMES IN COMPUTABLE INTEGRAL DOMAINS LEIGH EVRON, JOSEPH R. MILETI, AND ETHAN RATLIFF-CRAIN Abstract. A computable ring is a ring equipped with mechanical procedure to add and multiply

### Linear Algebra. Min Yan

Linear Algebra Min Yan January 2, 2018 2 Contents 1 Vector Space 7 1.1 Definition................................. 7 1.1.1 Axioms of Vector Space..................... 7 1.1.2 Consequence of Axiom......................

### Finite Fields. Sophie Huczynska. Semester 2, Academic Year

Finite Fields Sophie Huczynska Semester 2, Academic Year 2005-06 2 Chapter 1. Introduction Finite fields is a branch of mathematics which has come to the fore in the last 50 years due to its numerous applications,

### SUMMARY OF GROUPS AND RINGS GROUPS AND RINGS III Week 1 Lecture 1 Tuesday 3 March.

SUMMARY OF GROUPS AND RINGS GROUPS AND RINGS III 2009 Week 1 Lecture 1 Tuesday 3 March. 1. Introduction (Background from Algebra II) 1.1. Groups and Subgroups. Definition 1.1. A binary operation on a set

### CANONICAL FORMS FOR LINEAR TRANSFORMATIONS AND MATRICES. D. Katz

CANONICAL FORMS FOR LINEAR TRANSFORMATIONS AND MATRICES D. Katz The purpose of this note is to present the rational canonical form and Jordan canonical form theorems for my M790 class. Throughout, we fix

### Class Notes; Week 7, 2/26/2016

Class Notes; Week 7, 2/26/2016 Day 18 This Time Section 3.3 Isomorphism and Homomorphism [0], [2], [4] in Z 6 + 0 4 2 0 0 4 2 4 4 2 0 2 2 0 4 * 0 4 2 0 0 0 0 4 0 4 2 2 0 2 4 So {[0], [2], [4]} is a subring.

### A linear algebra proof of the fundamental theorem of algebra

A linear algebra proof of the fundamental theorem of algebra Andrés E. Caicedo May 18, 2010 Abstract We present a recent proof due to Harm Derksen, that any linear operator in a complex finite dimensional

### MT5836 Galois Theory MRQ

MT5836 Galois Theory MRQ May 3, 2017 Contents Introduction 3 Structure of the lecture course............................... 4 Recommended texts..................................... 4 1 Rings, Fields and

### Lagrange s polynomial

Lagrange s polynomial Nguyen Trung Tuan November 16, 2016 Abstract In this article, I will use Lagrange polynomial to solve some problems from Mathematical Olympiads. Contents 1 Lagrange s interpolation

### ALGEBRA II: RINGS AND MODULES OVER LITTLE RINGS.

ALGEBRA II: RINGS AND MODULES OVER LITTLE RINGS. KEVIN MCGERTY. 1. RINGS The central characters of this course are algebraic objects known as rings. A ring is any mathematical structure where you can add

### ALGEBRA QUALIFYING EXAM SPRING 2012

ALGEBRA QUALIFYING EXAM SPRING 2012 Work all of the problems. Justify the statements in your solutions by reference to specific results, as appropriate. Partial credit is awarded for partial solutions.

### A linear algebra proof of the fundamental theorem of algebra

A linear algebra proof of the fundamental theorem of algebra Andrés E. Caicedo May 18, 2010 Abstract We present a recent proof due to Harm Derksen, that any linear operator in a complex finite dimensional

### LECTURE NOTES IN CRYPTOGRAPHY

1 LECTURE NOTES IN CRYPTOGRAPHY Thomas Johansson 2005/2006 c Thomas Johansson 2006 2 Chapter 1 Abstract algebra and Number theory Before we start the treatment of cryptography we need to review some basic

### Generator Matrix. Theorem 6: If the generator polynomial g(x) of C has degree n-k then C is an [n,k]-cyclic code. If g(x) = a 0. a 1 a n k 1.

Cyclic Codes II Generator Matrix We would now like to consider how the ideas we have previously discussed for linear codes are interpreted in this polynomial version of cyclic codes. Theorem 6: If the

### 50 Algebraic Extensions

50 Algebraic Extensions Let E/K be a field extension and let a E be algebraic over K. Then there is a nonzero polynomial f in K[x] such that f(a) = 0. Hence the subset A = {f K[x]: f(a) = 0} of K[x] does

### Algebra for error control codes

Algebra for error control codes EE 387, Notes 5, Handout #7 EE 387 concentrates on block codes that are linear: Codewords components are linear combinations of message symbols. g 11 g 12 g 1n g 21 g 22

### Polynomials, Ideals, and Gröbner Bases

Polynomials, Ideals, and Gröbner Bases Notes by Bernd Sturmfels for the lecture on April 10, 2018, in the IMPRS Ringvorlesung Introduction to Nonlinear Algebra We fix a field K. Some examples of fields

### Homework 8 Solutions to Selected Problems

Homework 8 Solutions to Selected Problems June 7, 01 1 Chapter 17, Problem Let f(x D[x] and suppose f(x is reducible in D[x]. That is, there exist polynomials g(x and h(x in D[x] such that g(x and h(x

### Finite Fields and Error-Correcting Codes

Lecture Notes in Mathematics Finite Fields and Error-Correcting Codes Karl-Gustav Andersson (Lund University) (version 1.013-16 September 2015) Translated from Swedish by Sigmundur Gudmundsson Contents

### Note that a unit is unique: 1 = 11 = 1. Examples: Nonnegative integers under addition; all integers under multiplication.

Algebra fact sheet An algebraic structure (such as group, ring, field, etc.) is a set with some operations and distinguished elements (such as 0, 1) satisfying some axioms. This is a fact sheet with definitions

### Fields in Cryptography. Çetin Kaya Koç Winter / 30

Fields in Cryptography http://koclab.org Çetin Kaya Koç Winter 2017 1 / 30 Field Axioms Fields in Cryptography A field F consists of a set S and two operations which we will call addition and multiplication,

### 1. General Vector Spaces

1.1. Vector space axioms. 1. General Vector Spaces Definition 1.1. Let V be a nonempty set of objects on which the operations of addition and scalar multiplication are defined. By addition we mean a rule

### The Cyclic Decomposition Theorem

The Cyclic Decomposition Theorem Math 481/525, Fall 2009 Let V be a finite-dimensional F -vector space, and let T : V V be a linear transformation. In this note we prove that V is a direct sum of cyclic

### Serge Ballif January 18, 2008

ballif@math.psu.edu The Pennsylvania State University January 18, 2008 Outline Rings Division Rings Noncommutative Rings s Roots of Rings Definition A ring R is a set toger with two binary operations +

### GEOMETRIC CONSTRUCTIONS AND ALGEBRAIC FIELD EXTENSIONS

GEOMETRIC CONSTRUCTIONS AND ALGEBRAIC FIELD EXTENSIONS JENNY WANG Abstract. In this paper, we study field extensions obtained by polynomial rings and maximal ideals in order to determine whether solutions

### (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

### The definition of a vector space (V, +, )

The definition of a vector space (V, +, ) 1. For any u and v in V, u + v is also in V. 2. For any u and v in V, u + v = v + u. 3. For any u, v, w in V, u + ( v + w) = ( u + v) + w. 4. There is an element

### 1. 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

### Math 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)