2. Let p be a prime and let a € F,- {0}. (a) Show that there exists a natural number meN such that a that the set {a: kN) is finite.] = 1. [Hint: first show (b) The order of a, denoted ord(a), is defined to be the smallest natural number n such that a" = 1. What are the orders of the elements of F,- {0} for p = 3,5,7. (e) Show that if a" = 1 for some m € Z then n = ord(a) divides m. [Hint: show that ged(m, n) = n.] (d) Show that ord(a) divides p - 1. [Hint: you may use Fermat's Little Theorem from

2. Let p be a prime and let a € F,- {0}. (a) Show that there exists a natural number meN such that a that the set {a: kN) is finite.] = 1. [Hint: first show (b) The order of a, denoted ord(a), is defined to be the smallest natural number n such that a" = 1. What are the orders of the elements of F,- {0} for p = 3,5,7. (e) Show that if a" = 1 for some m € Z then n = ord(a) divides m. [Hint: show that ged(m, n) = n.] (d) Show that ord(a) divides p - 1. [Hint: you may use Fermat's Little Theorem from

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter2: The Integers

Section2.8: Introduction To Cryptography (optional)

Problem 23E

See similar textbooks