{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

Cosets and quotient group homomorphisms and

Info icon This preview shows pages 4–8. Sign up to view the full content.

View Full Document Right Arrow Icon
Subgroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 4.3 Cosets and Quotient Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 4.4 Group Homomorphisms and Isomorphisms . . . . . . . . . . . . . . . . . . . . . . . . 26 4.5 Cyclic Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 4.6 The Structure of Finite Abelian Groups . . . . . . . . . . . . . . . . . . . . . . . . . 32 5 Rings 35 5.1 Definitions, Basic Properties, and Examples . . . . . . . . . . . . . . . . . . . . . . . 35 5.2 Polynomial rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 5.3 Ideals and Quotient Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 5.4 Ring homomorphisms and isomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . 41 6 Polynomials over Fields 44 7 The Structure of Z * n 48 iii
Image of page 4

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
8 Computing Generators and Discrete Logarithms in Z * p 51 8.1 Finding a Generator for Z * p . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 8.2 Computing Discrete Logarithms Z * p . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 8.3 Further remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 9 Quadratic Residues and Quadratic Reciprocity 56 9.1 Quadratic Residues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 9.2 The Legendre Symbol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 9.3 The Jacobi Symbol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 10 Computational Problems Related to Quadratic Residues 61 10.1 Computing the Jacobi Symbol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 10.2 Testing quadratic residuosity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 10.3 Computing modular square roots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 11 Primality Testing 65 11.1 Trial Division . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 11.2 A Fast Probabilistic Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 11.3 The Distribution of Primes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 11.4 Deterministic Primality Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 iv
Image of page 5
Chapter 1 Basic Properties of the Integers This chapter reviews some of the basic properties of the integers, including notions of divisibility and primality, unique factorization into primes, greatest common divisors, and least common multiples. 1.1 Divisibility and Primality Consider the integers Z = { . . . , - 1 , 0 , 1 , 2 , . . . } . For a, b Z , we say that b divides a , and write b | a , if there exists c Z such that a = bc . If b | a , then b is called a divisor of a . If b does not divide a , then we write b - a . We first state some simple facts: Theorem 1.1 For all a, b, c Z , we have 1. a | a , 1 | a , and a | 0 ; 2. 0 | a if and only if a = 0 ; 3. a | b and b | c implies a | c ; 4. a | b implies a | bc ; 5. a | b and a | c implies a | b + c ; 6. a | b and b | a if and only if a = ± b . Proof. Exercise. 2 We say that an integer p is prime if p > 1 and the only divisors of p are ± 1 and ± p . Conversely, and integer n is called composite if n > 1 and it is not prime. So an integer n > 1 is composite if and only if n = ab for some integers a, b with 1 < a, b < n . A fundamental fact is that any integer can be written as a signed product of primes in an essentially unique way. More precisely: Theorem 1.2 Every non-zero integer n can be expressed as n = ± Y p p ν p ( n ) , where the product is over all primes, and all but a finite number of the exponents are zero. Moreover, the exponents and sign are uniquely determined by n . 1
Image of page 6

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
To do prove this theorem, we may clearly assume that n is positive. The proof of the existence part of Theorem 1.2 is easy. If n is 1 or prime, we are done; otherwise, there exist a, b Z with 1 < a, b < n and n = ab , and we apply an inductive argument with a and b . The proof of the uniqueness part of Theorem 1.2 is not so simple, and most of the rest of this chapter is devoted to developing the ideas behind such a proof. The essential ingredient in the proof is the following: Theorem 1.3 (Division with Remainder Property) For a, b Z with b > 0 , there exist unique q, r Z such that a = bq + r and 0 r < b .
Image of page 7
Image of page 8
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern