# sol5 - Math 465 Solution Set 5 1 Find all incongruent...

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Math 465 Solution Set 5 Ae Ja Yee 1. Find all incongruent quadratic residues and nonresidues modulo 13. Solution. We square the ﬁrst 6 integers: 1 2 1 , 2 2 4 , 3 3 9 , 4 2 3 , 5 2 12 , 6 2 10 . Since we know there are 6 quadratic residues, they are the incongruent quadratic residues mod 13. Thus the quadratic residues mod 13 are: 1 , 3 , 4 , 9 , 10 , 12 , and the quadratic nonresidues mod 13 are: 2 , 5 , 6 , 7 , 8 , 11 . 2. Use Euler’s Criterion to evaluate the following Legendre symbols: (a) ( 11 23 ) (b) ( - 6 11 ) Solution. (a) ± 11 23 ² 11 (23 - 1) / 2 11 11 ( - 3) 3 11 2 ≡ - 4 · 6 ≡ - 1 (mod 23) Thus ( 11 23 ) = - 1. (b) ± - 6 11 ² ( - 6) (11 - 1) / 2 ( - 6) 5 5 5 3 2 · 5 1 (mod 11) Thus ( - 6 11 ) = 1. 3. Use Gauss’s Lemma to evaluate the following Legendre symbols: (a) ( 12 23 ) (b) ( - 5 11 ) Solution. (a) Consider 12 , 24 , 36 , 48 , 60 , 72 , 84 , 96 , 108 , 120 , 132. Then 12 , 24 1 , 36 13 , 48 2 , 60 14 , 72

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sol5 - Math 465 Solution Set 5 1 Find all incongruent...

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