Let p ≥ 5 be a prime. Let g be a primitive root of p. - Let p≥ 5 be a prime. Let g be a primitive root of p.P(i). If g-¹ mod p is the modular inverse of g, prove that g-¹ is also a primitiveroot of p.(ii). Prove that g ‡ g-¹ (mod p). (Hint: Prove first that g = g-¹ (mod p)implies that g² = 1 (mod p).)(iii). Recall that there are (p(p)) (p-1) primitive roots of p among{1,2,..., p}. We denote them by 9₁, 92, ..., 96(p-1). Prove that=o(p-1)II 9₁ = 1 (mod p).i=1

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Let p≥ 5 be a prime. Let g be a primitive root of p. (i). If g-¹ mod p is the modular inverse of g, prove that g-¹ is also a primitiv root of p. (ii). Prove that g‡ g-¹ (mod p). (Hint: Prove first that g = g-¹ (mod p) implies that g² = 1 (mod p).) iii). Recall that there are ((p)) (p-1) primitive roots of p among {1,2,...,p}. We denote them by 9₁, 92, ..., 96(p-1). Prove that =

Let p≥ 5 be a prime. Let g be a primitive root of p. (i). If g-¹ mod p is the modular inverse of g, prove that g-¹ is also a primitiv root of p. (ii). Prove that g‡ g-¹ (mod p). (Hint: Prove first that g = g-¹ (mod p) implies that g² = 1 (mod p).) iii). Recall that there are ((p)) (p-1) primitive roots of p among {1,2,...,p}. We denote them by 9₁, 92, ..., 96(p-1). Prove that =

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter2: The Integers

Section2.5: Congruence Of Integers

Problem 54E: 54. Let be a prime integer. Prove Fermat's Little Theorem: For any positive integer,. (Hint: Use...

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