hw5 - p while if p 3(mod 4 then-a is a quadratic...

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Math 465 Problem Set 5 Due Friday, March 7, 2008 1. Find all quadratic residues and non-residues modulo 13. 2. Use Euler’s Criterion to evaluate the following Legendre symbols: (a) ( 11 23 ) (b) ( - 6 11 ) 3. Use Gauss’s Lemma to evaluate the following Legendre symbols: (a) ( 12 23 ) (b) ( - 5 11 ) 4. Let p be an odd prime and a be a quadratic residue of p . Prove that if p 1 (mod 4), then - a is also a quadratic residue of
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Unformatted text preview: p , while if p ≡ 3 (mod 4), then-a is a quadratic non-residue of p . 5. Prove that if p is an odd prime and a, b are integers with ( a, p ) = 1, then p X n =1 ± an + b p ² = 0 . (Hint: show that { an + b | ( a, p ) = 1 , n = 1 , 2 , . . . , p } is a complete residue system modulo p , and that ∑ p x =1 ³ x p ´ = 0.) 1...
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