# Some facts about polynomials modulo m (Full proof of the Fingerprinting Theorem)

Size: px
Start display at page:

Transcription

1 Some facts about polynomials modulo m (Full proof of the Fingerprinting Theorem) In order to understand the details of the Fingerprinting Theorem on fingerprints of different texts from Chapter 19 of the book Algorithms Unplugged [AU2011], you have to look at polynomials modulo m. For this you need a little patience, and you should not be afraid of a fumbling around with variables, unknowns, integers, and prime numbers. The award in the end is a full proof of the theorem. We have calculated a fingerprint for a text (a 1,..., a n ) (of numbers between 0 and m 1) as follows: (1) (a 1 r n + a 2 r n a n 1 r + a n r) mod m. Here r was a number between 1 and m 1. However, r = 0 is also allowed, it only always gives the result 0. Let us look a little more closely at expressions as in (1). Since we don t know in advance which r is to be substituted, we write a symbol x in place of r, a variable. In this way we enter the world of polynomials. Polynomials modulo m Let us look at m = 17 as an example. Then mod-m-polynomials are expressions like 10x x + 2 or x 3 + 2x or x This means there are powers x 0 = 1, x 1 = x, x 2, x 3, x 4,..., of a symbol x (the variable). Among these powers there is the expression x 0, to be read as 1 and to be omitted when it appears as a factor: 2 x 0 = 2. Instead of x 1 we write x. Such powers of x we can multiply with numbers c between 0 and m 1 to obtain terms cx j. The factor c here is then called a coefficient. We get polynomials by adding arbitrary terms with different powers of x. In order not to get confused we normally order the terms according to falling powers of x, but we may also write 5x + 3x or 5x + 2x x 2 : these are just a different way of writing 3x 2 + 5x

2 If we are given an expression like 2x 4 3x x that is not really a polynomial modulo 17, because coefficients that are negative or larger than 16 are illegal, we may just take all numbers modulo 17 to obtain a proper mod-17-polynomial: 2x x x The general format for a mod-m-polynomial is the following: (2) c n x n + c n 1 x n c 2 x 2 + c 1 x + c 0. Here c n,..., c 0 are numbers between 0 and m 1 (endpoints included). 1 It is interesting to note that if one wants to calculate with polynomials there is no need to deal with the x and its powers at all. One just stores the numbers c 0, c 1,..., c n, e. g. in an array C[0..n], and has the complete information. Arithmetic for mod-m-polynomials With mod-m-polynomials we can carry out the arithmetic operations addition, subtraction, and multiplication. This is very easy, if one just applies the standard rules for placing things outside brackets or multiplying out products of sums. With coefficients we always calculate modulo m. Addition: We add two mod-m-polynomials by calculating as if x was just some unknown and collect terms that have the same power of x in them. The coefficients in these terms are added of course modulo m. For example (m = 17): (2x 4 + 3x x + 3) + (3x x x + 4) = 3x 5 + 2x 4 + 3x + 7. (The summand with x 3 disappears, because (3 + 14) mod 17 = 0.) Subtraction: We proceed in a way very similar to addition. For example: (10x 3 + 3x + 2) (x 3 + 2x + 13) = 9x 3 + x In comparison to (1) the numbering of the terms is reversed and a summand c 0 has been added. 2

3 We subtract the coefficients that come with the same powers of x, always thinking modulo m. One can avoid subtraction or the use of negative numbers by first replacing coefficient c by m c in the polynomial to be subtracted, and add the resulting polynomial. In the example from above the calculation then runs as follows: (10x 3 + 3x + 2) + (16x x + 4) = 9x 3 + x + 6. A special situation occurs if one subtracts a polynomial from itself: (10x 3 + 3x + 2) (10x 3 + 3x + 2) = (10x 3 + 3x + 2) + (7x x + 15) = 0. The result is a polynomial in which all powers of x have a coefficient that is 0. This polynomial is called the zero polynomial. Although normally one would not really like to deal with such a strange thing that tells us nothing, we will see that the zero polynomial is very special and it is very important for our purpose that we recognize it when it shows up. Multiplication: In order to multiply two polynomials (modulo m) we multiply out and then collect terms with the same power of x. For example: (3) (10x 4 + 3x + 2) (x 3 + 2x + 13) = (10x 7 + 3x x 4 ) + (3x 4 + 6x 2 + 5x) + (2x 3 + 4x + 9) = 10x 7 + 3x x 4 + 2x 3 + 6x 2 + 9x + 9. Is there also a division for polynomials? Yes, but that is a little more complicated. Division: After the result of the multiplication from (3) we will certainly write (always modulo m): (10x 7 + 3x x 4 + 2x 3 + 6x 2 + 9x + 9) : (x 3 + 2x + 13) = 10x 4 + 3x + 2. Somtimes however (well, actually more often than not) it doesn t add up. Just as with ordinary numbers we then do division with remainder. We don t want to run into problems with dividing numbers modulo m, so as divisors we admit only polynomials in which the highest power of x appears with a coefficient of 1. For example, x is admissible as divisor, but 10x 2 + 3x + 1 is not. The remainder in polynomial division is again a polynomial whose highest power of x is smaller than the highest power of x that occurs in the divisor. 3

4 Exampe: We have (modulo 17) and so we write (2x + 5) (x 2 + 4) + (2x + 4) = 2x 3 + 5x x + 7, (4) (2x 3 + 5x x + 7) : (x 2 + 4) = (x 2 + 4) with remainder 2x + 4. There is even an algorithm for dividing a polynomial f by a polynomial g, which always calculates the quotient polynomial and the remainder polynomial. We don t need such an algorithm here. All we have to know is that for every polynomial f and every polynomial g that has coefficient 1 with its highest power of x (which of course is at least x 0 ) there is a quotient polynomial q and a remainder polynomial re so that the following equation holds: (5) f = g q + re ; here the highest power of x appearing in re is smaller than the highest power of x in g. For example, in (4) the highest power of x in the remainder x = x 1 is smaller than the highest power of x in the divisor (x 2 ). For our fingerprint application we only need an extremely simple case: Division by polynomials x + s, for s a number. With a little patience you can check that (here: s = 3): That is, 3x x 3 + 5x 2 + 2x + 6 = (3x 3 + 4x x + 6) (x + 3) + 5. (3x x 3 + 5x 2 + 2x + 6) : (x + 3) = 3x 3 + 4x x + 6 with remainder 5. In general it is clear that when dividing by a polynomial x + s the remainder is simply a number. 2 2 The interested reader may want to try and find an algorithm to divide a polynomial c n x n + + c 1 x + c 0 by a polynomial x + s. 4

5 Substituting into polynomials, roots, factoring out linear factors x r Up to here we only calculated with polynomials as formal expressions, never touching the variable x. Another very important operation with polynomials is substitution. That means, at least in our simple situation, that we replace the x by some number between 0 and m 1 and evaluate to see what the resulting number is. For example, if in the polynomial f = 3x x 3 + 5x 2 + 2x + 1 we substitute the number 2 for x, we get f(2) = ( ) mod 17 = 7, if we substitute the number 14 for x, we get and so on. f(14) = ( ) mod 17 = 0, If f(r) = 0 we call r a root of f. Thus, r = 14 is a root of f = 3x x 3 + 5x 2 + 2x + 1. Now we divide f by x 14 = x + 3 (viewed modulo 17) and note: (3x x 3 + 5x 2 + 2x + 1) : (x + 3) = 3x 3 + 4x x + 6. There is no remainder in the division! We can try more examples with dividing polynomials by x r, where r is a root, and will always find that there is no remainder. Here s the general fact, and because it is very important for us, we prove it. Theorem 1 For polynomials modulo m, for a number m 2, we have: if r is a root of the polynomial f, then division of f by x r ( = x + (m r)) yields remainder 0. This means: It is possible to write f = (x r) q, for some (quotient) polynomial q. 5

6 Proof : We can certainly divide f by x r, with remainder, that is, we can write: f = q (x r) + re. Here re is a number between 0 and m 1, and q is a polynomial. Now we substitute r for x on both sides. Because r is a root of f, we get: 0 = f(r) = q(r) (r r) + re = re. Since the remainder re is 0, we get f = q (x r), and Theorem 1 is proved. One can put Theorem 1 as follows: If r is a root of f, one can factor out the factor x r from f. The next theorem says that if m is a prime number this can be done with several roots of a polynomial one after the other. The proof is a little more tricky. Theorem 2 Consider polynomials modulo m, for a prime number m 2. If k 1 and r 1,..., r k are different roots of f between 0 and m 1, then it is possible to write for some polynomial h. f = (x r 1 ) (x r 2 ) (x r k ) h, Proof : Let us first consider the case where f has two different roots r und t. By Theorem 1 we can write f = q (x r). Since t is a root of f, we have (modulo m): 0 = f(t) = q(t) (t r). This means: The number q(t) (t r), calculated with integer arithmetic (without taking remainders modulo m), is divisible by m. Because t and r are different numbers between 0 and m 1, the number t r is not divisible by m. Now it is a well known basic fact about prime numbers m that from m is a divisor of a b 6

7 it follows that m must divide a or b (or both) 3. This gets us that m divides q(t) (in the integers). Viewed modulo m this means that q(t) = 0. Thus t is a root of the quotient polynomial q. We apply Theorem 1 once again and write q = p (x t) for a suitable polynomial p. Thus we get that f = p (x t) (x r), as desired. Example: For m = 17 and (6) f = x x 3 + x x + 11 we have that f(2) = f(16) = 0 and that f = (x 2) (x 16) (x 2 + 3) = (x + 15) (x + 1) (x 2 + 3). If f happens to have more than two roots, one can proceed along the same pattern and keep factoring out linear factors, until the desired representation f = (x r 1 ) (x r 2 ) (x r k ) h is obtained. (Formally, one uses induction on k.) Now we can formulate and prove the statement we need for our purposes. Theorem 3 Consider polynomials modulo m, for a prime number m 2. If f is a polynomial that is not the zero polynomial, and x n is the highest power of x that occurs in f (with a coefficient c n 0), then f has no more than n distinct roots between 0 and m 1. Examples: (Always modulo 17.) The polynomial 2x + 11 has one root, which is 3; the polynomial 2x 2 + 7x + 12 has two roots, which are 2 and 3; the polynomial x x 3 + x x + 11 from (6) cannot have more than four roots (actually, there are only two roots: 2 and 16). Proof of Theorem 3: We take any k distinct roots of f and call them r 1,..., r k. The aim is to show that k cannot be larger than n. By Theorem 2 we can write: (7) f = (x r 1 ) (x r 2 ) (x r k ) h. 3 For numbers m that are not prime this is not necessarily so: the number 6 divides 4 9 = 36, but 6 divides neither 4 nor 9. 7

8 Here h cannot be the zero polynomial, since otherwise f would be the zero polynomial. In h there is a highest power of x, dx l, say, with d 0, l 0. Now if we multiply out the right hand side of (7) we obtain the power x k+l, with factor d as well, and there is no other term in the product with a larger power of x. Since x n is the highest power of x in f with a nonzero coefficient, we must have n = k + l k, as desired. Polynomials and fingerprints After these preparations we can finally prove the fingerprinting theorem from [AU2011]. Fingerprinting Theorem If T A and T B are different texts of length n, and if m is a prime number that is larger than the largest number occurring in T A and T B, then at most n out of the m pairs of numbers can consist of two equal numbers. FP(T A, r), FP(T B, r), 0 r < m, Proof : We look at two texts (that is to say: sequences of numbers) T A = (a 1,..., a n ) and T B = (b 1,..., b n ). The numbers a 1,..., a n, b 1,..., b n are between 0 and m 1. From these texts we form two polynomials: f A = a 1 x n + + a n x and f B = b 1 x n + + b n x, and the polynomial g obtained from f A and f B by subtraction modulo m: g = f A f B = (a 1 b 1 ) x n + + (a n b n ) x. Now if T A and T B are different texts, then (also if we look at the numbers modulo m) at least one of the coefficients (a i b i ) in g is not zero, hence g is not the zero polynomial. Obviously, in g no power of x with an exponent larger than n can occur. (Note that x n need not occur, since we could have 8

9 a n = b n. It very much depends on the texts what g looks like. Maybe it has only one nonzero term.) Now, obviously, FP(T A, r) = f A (r) and FP(T B, r) = f B (r). So if the fingerprints FP(T A, r) and FP(T B, r) are equal, then f A (r) = f B (r), hence g(r) = 0 and this means that r is a root of g. By Theorem 3 the polynomial g cannot have more than n roots. Hence there are no more than n numbers r with FP(T A, r) = FP(T B, r), and hence the Fingerprinting Theorem is proved. Literatur [AU2011] Vöcking, B., Alt, H., Dietzfelbinger, M., Reischuk, R., Scheideler, C., Vollmer, H., Wagner, D. (Eds.), Algorithms Unplugged, Springer- Verlag, 2011, ISBN: c Martin Dietzfelbinger

### Zeros of Polynomial Functions

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

### CHAPTER SIX IRREDUCIBILITY AND FACTORIZATION 1. BASIC DIVISIBILITY THEORY

January 10, 2010 CHAPTER SIX IRREDUCIBILITY AND FACTORIZATION 1. BASIC DIVISIBILITY THEORY The set of polynomials over a field F is a ring, whose structure shares with the ring of integers many characteristics.

### 3.2 The Factor Theorem and The Remainder Theorem

3. The Factor Theorem and The Remainder Theorem 57 3. The Factor Theorem and The Remainder Theorem Suppose we wish to find the zeros of f(x) = x 3 + 4x 5x 4. Setting f(x) = 0 results in the polynomial

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

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

### The Factor Theorem and a corollary of the Fundamental Theorem of Algebra

Math 421 Fall 2010 The Factor Theorem and a corollary of the Fundamental Theorem of Algebra 27 August 2010 Copyright 2006 2010 by Murray Eisenberg. All rights reserved. Prerequisites Mathematica Aside

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

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

### JUST THE MATHS UNIT NUMBER 1.8. ALGEBRA 8 (Polynomials) A.J.Hobson

JUST THE MATHS UNIT NUMBER 1.8 ALGEBRA 8 (Polynomials) by A.J.Hobson 1.8.1 The factor theorem 1.8.2 Application to quadratic and cubic expressions 1.8.3 Cubic equations 1.8.4 Long division of polynomials

### Copy 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,...}

### Revised Version of Chapter 23. We learned long ago how to solve linear congruences. ax c (mod m)

Chapter 23 Squares Modulo p Revised Version of Chapter 23 We learned long ago how to solve linear congruences ax c (mod m) (see Chapter 8). It s now time to take the plunge and move on to quadratic equations.

### SOLVING POLYNOMIAL EQUATIONS

C SOLVING POLYNOMIAL EQUATIONS We will assume in this appendix that you know how to divide polynomials using long division and synthetic division. If you need to review those techniques, refer to an algebra

### Quotient 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.

### FACTORING POLYNOMIALS IN THE RING OF FORMAL POWER SERIES OVER Z

FACTORING POLYNOMIALS IN THE RING OF FORMAL POWER SERIES OVER Z DANIEL BIRMAJER, JUAN B GIL, AND MICHAEL WEINER Abstract We consider polynomials with integer coefficients and discuss their factorization

### Continued Fractions and the Euclidean Algorithm

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

### Homework until Test #2

MATH31: Number Theory Homework until Test # Philipp BRAUN Section 3.1 page 43, 1. It has been conjectured that there are infinitely many primes of the form n. Exhibit five such primes. Solution. Five such

### 8 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.

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

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

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

### Zero: If P is a polynomial and if c is a number such that P (c) = 0 then c is a zero of P.

MATH 11011 FINDING REAL ZEROS KSU OF A POLYNOMIAL Definitions: Polynomial: is a function of the form P (x) = a n x n + a n 1 x n 1 + + a x + a 1 x + a 0. The numbers a n, a n 1,..., a 1, a 0 are called

### Math Review. for the Quantitative Reasoning Measure of the GRE revised General Test

Math Review for the Quantitative Reasoning Measure of the GRE revised General Test www.ets.org Overview This Math Review will familiarize you with the mathematical skills and concepts that are important

### Integer roots of quadratic and cubic polynomials with integer coefficients

Integer roots of quadratic and cubic polynomials with integer coefficients Konstantine Zelator Mathematics, Computer Science and Statistics 212 Ben Franklin Hall Bloomsburg University 400 East Second Street

### The Division Algorithm for Polynomials Handout Monday March 5, 2012

The Division Algorithm for Polynomials Handout Monday March 5, 0 Let F be a field (such as R, Q, C, or F p for some prime p. This will allow us to divide by any nonzero scalar. (For some of the following,

### 9. POLYNOMIALS. Example 1: The expression a(x) = x 3 4x 2 + 7x 11 is a polynomial in x. The coefficients of a(x) are the numbers 1, 4, 7, 11.

9. POLYNOMIALS 9.1. Definition of a Polynomial A polynomial is an expression of the form: a(x) = a n x n + a n-1 x n-1 +... + a 1 x + a 0. The symbol x is called an indeterminate and simply plays the role

### The finite field with 2 elements The simplest finite field is

The finite field with 2 elements The simplest finite field is GF (2) = F 2 = {0, 1} = Z/2 It has addition and multiplication + and defined to be 0 + 0 = 0 0 + 1 = 1 1 + 0 = 1 1 + 1 = 0 0 0 = 0 0 1 = 0

### a 1 x + a 0 =0. (3) ax 2 + bx + c =0. (4)

ROOTS OF POLYNOMIAL EQUATIONS In this unit we discuss polynomial equations. A polynomial in x of degree n, where n 0 is an integer, is an expression of the form P n (x) =a n x n + a n 1 x n 1 + + a 1 x

### Zeros of Polynomial Functions

Zeros of Polynomial Functions The Rational Zero Theorem If f (x) = a n x n + a n-1 x n-1 + + a 1 x + a 0 has integer coefficients and p/q (where p/q is reduced) is a rational zero, then p is a factor of

### = 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

### CHAPTER 5. Number Theory. 1. Integers and Division. Discussion

CHAPTER 5 Number Theory 1. Integers and Division 1.1. Divisibility. Definition 1.1.1. Given two integers a and b we say a divides b if there is an integer c such that b = ac. If a divides b, we write a

### Notes on Factoring. MA 206 Kurt Bryan

The General Approach Notes on Factoring MA 26 Kurt Bryan Suppose I hand you n, a 2 digit integer and tell you that n is composite, with smallest prime factor around 5 digits. Finding a nontrivial factor

### Polynomial and Synthetic Division. Long Division of Polynomials. Example 1. 6x 2 7x 2 x 2) 19x 2 16x 4 6x3 12x 2 7x 2 16x 7x 2 14x. 2x 4.

_.qd /7/5 9: AM Page 5 Section.. Polynomial and Synthetic Division 5 Polynomial and Synthetic Division What you should learn Use long division to divide polynomials by other polynomials. Use synthetic

### 63. Graph y 1 2 x and y 2 THE FACTOR THEOREM. The Factor Theorem. Consider the polynomial function. P(x) x 2 2x 15.

9.4 (9-27) 517 Gear ratio d) For a fixed wheel size and chain ring, does the gear ratio increase or decrease as the number of teeth on the cog increases? decreases 100 80 60 40 20 27-in. wheel, 44 teeth

### SECTION 0.6: POLYNOMIAL, RATIONAL, AND ALGEBRAIC EXPRESSIONS

(Section 0.6: Polynomial, Rational, and Algebraic Expressions) 0.6.1 SECTION 0.6: POLYNOMIAL, RATIONAL, AND ALGEBRAIC EXPRESSIONS LEARNING OBJECTIVES Be able to identify polynomial, rational, and algebraic

### 8 Divisibility and prime numbers

8 Divisibility and prime numbers 8.1 Divisibility In this short section we extend the concept of a multiple from the natural numbers to the integers. We also summarize several other terms that express

### Factoring 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

### a 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

### Jacobi s four squares identity Martin Klazar

Jacobi s four squares identity Martin Klazar (lecture on the 7-th PhD conference) Ostrava, September 10, 013 C. Jacobi [] in 189 proved that for any integer n 1, r (n) = #{(x 1, x, x 3, x ) Z ( i=1 x i

### UNCORRECTED PAGE PROOFS

number and and algebra TopIC 17 Polynomials 17.1 Overview Why learn this? Just as number is learned in stages, so too are graphs. You have been building your knowledge of graphs and functions over time.

### POLYNOMIAL FUNCTIONS

POLYNOMIAL FUNCTIONS Polynomial Division.. 314 The Rational Zero Test.....317 Descarte s Rule of Signs... 319 The Remainder Theorem.....31 Finding all Zeros of a Polynomial Function.......33 Writing a

### SECTION 2.5: FINDING ZEROS OF POLYNOMIAL FUNCTIONS

SECTION 2.5: FINDING ZEROS OF POLYNOMIAL FUNCTIONS Assume f ( x) is a nonconstant polynomial with real coefficients written in standard form. PART A: TECHNIQUES WE HAVE ALREADY SEEN Refer to: Notes 1.31

### Gröbner Bases and their Applications

Gröbner Bases and their Applications Kaitlyn Moran July 30, 2008 1 Introduction We know from the Hilbert Basis Theorem that any ideal in a polynomial ring over a field is finitely generated [3]. However,

### GREATEST COMMON DIVISOR

DEFINITION: GREATEST COMMON DIVISOR The greatest common divisor (gcd) of a and b, denoted by (a, b), is the largest common divisor of integers a and b. THEOREM: If a and b are nonzero integers, then their

### Partial Fractions. (x 1)(x 2 + 1)

Partial Fractions Adding rational functions involves finding a common denominator, rewriting each fraction so that it has that denominator, then adding. For example, 3x x 1 3x(x 1) (x + 1)(x 1) + 1(x +

### Partial Fractions. Combining fractions over a common denominator is a familiar operation from algebra:

Partial Fractions Combining fractions over a common denominator is a familiar operation from algebra: From the standpoint of integration, the left side of Equation 1 would be much easier to work with than

### 2.3. Finding polynomial functions. An Introduction:

2.3. Finding polynomial functions. An Introduction: As is usually the case when learning a new concept in mathematics, the new concept is the reverse of the previous one. Remember how you first learned

### Introduction to Finite Fields (cont.)

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

### FACTORING AFTER DEDEKIND

FACTORING AFTER DEDEKIND KEITH CONRAD Let K be a number field and p be a prime number. When we factor (p) = po K into prime ideals, say (p) = p e 1 1 peg g, we refer to the data of the e i s, the exponents

### Math 0980 Chapter Objectives. Chapter 1: Introduction to Algebra: The Integers.

Math 0980 Chapter Objectives Chapter 1: Introduction to Algebra: The Integers. 1. Identify the place value of a digit. 2. Write a number in words or digits. 3. Write positive and negative numbers used

### minimal polyonomial Example

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

### THE NUMBER OF REPRESENTATIONS OF n OF THE FORM n = x 2 2 y, x > 0, y 0

THE NUMBER OF REPRESENTATIONS OF n OF THE FORM n = x 2 2 y, x > 0, y 0 RICHARD J. MATHAR Abstract. We count solutions to the Ramanujan-Nagell equation 2 y +n = x 2 for fixed positive n. The computational

### FACTORING 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,

### Number Theory. Proof. Suppose otherwise. Then there would be a finite number n of primes, which we may

Number Theory Divisibility and Primes Definition. If a and b are integers and there is some integer c such that a = b c, then we say that b divides a or is a factor or divisor of a and write b a. Definition

### 3.1. RATIONAL EXPRESSIONS

3.1. RATIONAL EXPRESSIONS RATIONAL NUMBERS In previous courses you have learned how to operate (do addition, subtraction, multiplication, and division) on rational numbers (fractions). Rational numbers

### CS 103X: Discrete Structures Homework Assignment 3 Solutions

CS 103X: Discrete Structures Homework Assignment 3 s Exercise 1 (20 points). On well-ordering and induction: (a) Prove the induction principle from the well-ordering principle. (b) Prove the well-ordering

### Determinants can be used to solve a linear system of equations using Cramer s Rule.

2.6.2 Cramer s Rule Determinants can be used to solve a linear system of equations using Cramer s Rule. Cramer s Rule for Two Equations in Two Variables Given the system This system has the unique solution

### Section 4.2: The Division Algorithm and Greatest Common Divisors

Section 4.2: The Division Algorithm and Greatest Common Divisors The Division Algorithm The Division Algorithm is merely long division restated as an equation. For example, the division 29 r. 20 32 948

### Settling a Question about Pythagorean Triples

Settling a Question about Pythagorean Triples TOM VERHOEFF Department of Mathematics and Computing Science Eindhoven University of Technology P.O. Box 513, 5600 MB Eindhoven, The Netherlands E-Mail address:

### Math 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

### March 29, 2011. 171S4.4 Theorems about Zeros of Polynomial Functions

MAT 171 Precalculus Algebra Dr. Claude Moore Cape Fear Community College CHAPTER 4: Polynomial and Rational Functions 4.1 Polynomial Functions and Models 4.2 Graphing Polynomial Functions 4.3 Polynomial

### 7. Some irreducible polynomials

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

### A Second Course in Mathematics Concepts for Elementary Teachers: Theory, Problems, and Solutions

A Second Course in Mathematics Concepts for Elementary Teachers: Theory, Problems, and Solutions Marcel B. Finan Arkansas Tech University c All Rights Reserved First Draft February 8, 2006 1 Contents 25

### Application. Outline. 3-1 Polynomial Functions 3-2 Finding Rational Zeros of. Polynomial. 3-3 Approximating Real Zeros of.

Polynomial and Rational Functions Outline 3-1 Polynomial Functions 3-2 Finding Rational Zeros of Polynomials 3-3 Approximating Real Zeros of Polynomials 3-4 Rational Functions Chapter 3 Group Activity:

### Zeros of a Polynomial Function

Zeros of a Polynomial Function An important consequence of the Factor Theorem is that finding the zeros of a polynomial is really the same thing as factoring it into linear factors. In this section we

### DIVISIBILITY AND GREATEST COMMON DIVISORS

DIVISIBILITY AND GREATEST COMMON DIVISORS KEITH CONRAD 1 Introduction We will begin with a review of divisibility among integers, mostly to set some notation and to indicate its properties Then we will

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

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

### Stupid Divisibility Tricks

Stupid Divisibility Tricks 101 Ways to Stupefy Your Friends Appeared in Math Horizons November, 2006 Marc Renault Shippensburg University Mathematics Department 1871 Old Main Road Shippensburg, PA 17013

### Discrete Mathematics: Homework 7 solution. Due: 2011.6.03

EE 2060 Discrete Mathematics spring 2011 Discrete Mathematics: Homework 7 solution Due: 2011.6.03 1. Let a n = 2 n + 5 3 n for n = 0, 1, 2,... (a) (2%) Find a 0, a 1, a 2, a 3 and a 4. (b) (2%) Show that

### 3 Monomial orders and the division algorithm

3 Monomial orders and the division algorithm We address the problem of deciding which term of a polynomial is the leading term. We noted above that, unlike in the univariate case, the total degree does

### SOLUTIONS FOR PROBLEM SET 2

SOLUTIONS FOR PROBLEM SET 2 A: There exist primes p such that p+6k is also prime for k = 1,2 and 3. One such prime is p = 11. Another such prime is p = 41. Prove that there exists exactly one prime p such

### HOMEWORK 5 SOLUTIONS. n!f n (1) lim. ln x n! + xn x. 1 = G n 1 (x). (2) k + 1 n. (n 1)!

Math 7 Fall 205 HOMEWORK 5 SOLUTIONS Problem. 2008 B2 Let F 0 x = ln x. For n 0 and x > 0, let F n+ x = 0 F ntdt. Evaluate n!f n lim n ln n. By directly computing F n x for small n s, we obtain the following

### z 0 and y even had the form

Gaussian Integers The concepts of divisibility, primality and factoring are actually more general than the discussion so far. For the moment, we have been working in the integers, which we denote by Z

### BEGINNING ALGEBRA ACKNOWLEDMENTS

BEGINNING ALGEBRA The Nursing Department of Labouré College requested the Department of Academic Planning and Support Services to help with mathematics preparatory materials for its Bachelor of Science

### it is easy to see that α = a

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

### Applications 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

### Polynomials. Dr. philippe B. laval Kennesaw State University. April 3, 2005

Polynomials Dr. philippe B. laval Kennesaw State University April 3, 2005 Abstract Handout on polynomials. The following topics are covered: Polynomial Functions End behavior Extrema Polynomial Division

### Lecture 6: Finite Fields (PART 3) PART 3: Polynomial Arithmetic. Theoretical Underpinnings of Modern Cryptography

Lecture 6: Finite Fields (PART 3) PART 3: Polynomial Arithmetic Theoretical Underpinnings of Modern Cryptography Lecture Notes on Computer and Network Security by Avi Kak (kak@purdue.edu) January 29, 2015

### 1 Lecture: Integration of rational functions by decomposition

Lecture: Integration of rational functions by decomposition into partial fractions Recognize and integrate basic rational functions, except when the denominator is a power of an irreducible quadratic.

### Copyrighted Material. Chapter 1 DEGREE OF A CURVE

Chapter 1 DEGREE OF A CURVE Road Map The idea of degree is a fundamental concept, which will take us several chapters to explore in depth. We begin by explaining what an algebraic curve is, and offer two

### The Euclidean Algorithm

The Euclidean Algorithm A METHOD FOR FINDING THE GREATEST COMMON DIVISOR FOR TWO LARGE NUMBERS To be successful using this method you have got to know how to divide. If this is something that you have

### Number Theory Hungarian Style. Cameron Byerley s interpretation of Csaba Szabó s lectures

Number Theory Hungarian Style Cameron Byerley s interpretation of Csaba Szabó s lectures August 20, 2005 2 0.1 introduction Number theory is a beautiful subject and even cooler when you learn about it

### ON THE EMBEDDING OF BIRKHOFF-WITT RINGS IN QUOTIENT FIELDS

ON THE EMBEDDING OF BIRKHOFF-WITT RINGS IN QUOTIENT FIELDS DOV TAMARI 1. Introduction and general idea. In this note a natural problem arising in connection with so-called Birkhoff-Witt algebras of Lie

### Lecture Notes on Polynomials

Lecture Notes on Polynomials Arne Jensen Department of Mathematical Sciences Aalborg University c 008 Introduction These lecture notes give a very short introduction to polynomials with real and complex

### Lagrange Interpolation is a method of fitting an equation to a set of points that functions well when there are few points given.

Polynomials (Ch.1) Study Guide by BS, JL, AZ, CC, SH, HL Lagrange Interpolation is a method of fitting an equation to a set of points that functions well when there are few points given. Sasha s method

### If A is divided by B the result is 2/3. If B is divided by C the result is 4/7. What is the result if A is divided by C?

Problem 3 If A is divided by B the result is 2/3. If B is divided by C the result is 4/7. What is the result if A is divided by C? Suggested Questions to ask students about Problem 3 The key to this question

### Breaking The Code. Ryan Lowe. Ryan Lowe is currently a Ball State senior with a double major in Computer Science and Mathematics and

Breaking The Code Ryan Lowe Ryan Lowe is currently a Ball State senior with a double major in Computer Science and Mathematics and a minor in Applied Physics. As a sophomore, he took an independent study

### Second Order Linear Nonhomogeneous Differential Equations; Method of Undetermined Coefficients. y + p(t) y + q(t) y = g(t), g(t) 0.

Second Order Linear Nonhomogeneous Differential Equations; Method of Undetermined Coefficients We will now turn our attention to nonhomogeneous second order linear equations, equations with the standard

### PROBLEM SET 6: POLYNOMIALS

PROBLEM SET 6: POLYNOMIALS 1. introduction In this problem set we will consider polynomials with coefficients in K, where K is the real numbers R, the complex numbers C, the rational numbers Q or any other

Chapter 2 Remodulization of Congruences Proceedings NCUR VI. è1992è, Vol. II, pp. 1036í1041. Jeærey F. Gold Department of Mathematics, Department of Physics University of Utah Don H. Tucker Department

### 4.3 Lagrange Approximation

206 CHAP. 4 INTERPOLATION AND POLYNOMIAL APPROXIMATION Lagrange Polynomial Approximation 4.3 Lagrange Approximation Interpolation means to estimate a missing function value by taking a weighted average

### South Carolina College- and Career-Ready (SCCCR) Algebra 1

South Carolina College- and Career-Ready (SCCCR) Algebra 1 South Carolina College- and Career-Ready Mathematical Process Standards The South Carolina College- and Career-Ready (SCCCR) Mathematical Process

### Indiana State Core Curriculum Standards updated 2009 Algebra I

Indiana State Core Curriculum Standards updated 2009 Algebra I Strand Description Boardworks High School Algebra presentations Operations With Real Numbers Linear Equations and A1.1 Students simplify and

### CORRELATED TO THE SOUTH CAROLINA COLLEGE AND CAREER-READY FOUNDATIONS IN ALGEBRA

We Can Early Learning Curriculum PreK Grades 8 12 INSIDE ALGEBRA, GRADES 8 12 CORRELATED TO THE SOUTH CAROLINA COLLEGE AND CAREER-READY FOUNDATIONS IN ALGEBRA April 2016 www.voyagersopris.com Mathematical

parent ROADMAP MATHEMATICS SUPPORTING YOUR CHILD IN HIGH SCHOOL HS America s schools are working to provide higher quality instruction than ever before. The way we taught students in the past simply does

### Cryptography and Network Security. Prof. D. Mukhopadhyay. Department of Computer Science and Engineering. Indian Institute of Technology, Kharagpur

Cryptography and Network Security Prof. D. Mukhopadhyay Department of Computer Science and Engineering Indian Institute of Technology, Kharagpur Module No. # 01 Lecture No. # 12 Block Cipher Standards

### Alum Rock Elementary Union School District Algebra I Study Guide for Benchmark III

Alum Rock Elementary Union School District Algebra I Study Guide for Benchmark III Name Date Adding and Subtracting Polynomials Algebra Standard 10.0 A polynomial is a sum of one ore more monomials. Polynomial

### SUNY ECC. ACCUPLACER Preparation Workshop. Algebra Skills

SUNY ECC ACCUPLACER Preparation Workshop Algebra Skills Gail A. Butler Ph.D. Evaluating Algebraic Epressions Substitute the value (#) in place of the letter (variable). Follow order of operations!!! E)

### Algebra Unpacked Content For the new Common Core standards that will be effective in all North Carolina schools in the 2012-13 school year.

This document is designed to help North Carolina educators teach the Common Core (Standard Course of Study). NCDPI staff are continually updating and improving these tools to better serve teachers. Algebra

### Unit 6: Polynomials. 1 Polynomial Functions and End Behavior. 2 Polynomials and Linear Factors. 3 Dividing Polynomials

Date Period Unit 6: Polynomials DAY TOPIC 1 Polynomial Functions and End Behavior Polynomials and Linear Factors 3 Dividing Polynomials 4 Synthetic Division and the Remainder Theorem 5 Solving Polynomial

### k, then n = p2α 1 1 pα k

Powers of Integers An integer n is a perfect square if n = m for some integer m. Taking into account the prime factorization, if m = p α 1 1 pα k k, then n = pα 1 1 p α k k. That is, n is a perfect square