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Question 5. Let p= 3 (mod 4) be a prime. Assume that a is an integer such that the order of a is (p-1)/2, i.e., a(p-¹)/2 = 1 (mod p) but a" # 1 (mod p) for any n with 1 ≤ n < (p-1)/2. Prove that – a is a primitive root modulo p.

Question 5. Let p= 3 (mod 4) be a prime. Assume that a is an integer such that the order of a is (p-1)/2, i.e., a(p-¹)/2 = 1 (mod p) but a" # 1 (mod p) for any n with 1 ≤ n < (p-1)/2. Prove that – a is a primitive root modulo p.

Elements Of Modern Algebra
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 30E: 30. Prove that any positive integer is congruent to its units digit modulo .
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Question 5. Let p = 3 (mod 4) be a prime. Assume that a is an integer such that the order of a is (p-1)/2, i.e., a(p-¹)/2 = 1 (mod p) but aª ‡ 1 (mod p) for any n with 1 ≤ n < (p−1)/2. Prove that –a is a primitive root modulo p.
Transcribed Image Text:Question 5. Let p = 3 (mod 4) be a prime. Assume that a is an integer such that the order of a is (p-1)/2, i.e., a(p-¹)/2 = 1 (mod p) but aª ‡ 1 (mod p) for any n with 1 ≤ n < (p−1)/2. Prove that –a is a primitive root modulo p.
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