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Let p be an odd prime and a be a quadratic residue of p. Prove that if p = ±1 (mod 8), then 2a is also a quadratic residue of p; if p = ±3 (mod 8), then 2a is a quadratic nonresidue of p.

Let p be an odd prime and a be a quadratic residue of p. Prove that if p = ±1 (mod 8), then 2a is also a quadratic residue of p; if p = ±3 (mod 8), then 2a is a quadratic nonresidue of p.

Elements Of Modern Algebra
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter2: The Integers
Section2.4: Prime Factors And Greatest Common Divisor
Problem 28E: Let and be positive integers. If and is the least common multiple of and , prove that . Note...
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Let p be an odd prime and a be a quadratic residue of p. Prove that if p = ±1 (mod 8), then 2a is also a quadratic residue of p; if p = ±3 (mod 8), then 2a is a quadratic nonresidue of p.
Transcribed Image Text:Let p be an odd prime and a be a quadratic residue of p. Prove that if p = ±1 (mod 8), then 2a is also a quadratic residue of p; if p = ±3 (mod 8), then 2a is a quadratic nonresidue of p.
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