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2. Use Gauss' Lemma, to compute (²) = (-1)², μ = |aPn N, -6 (30). 31

2. Use Gauss' Lemma, to compute (²) = (-1)², μ = |aPn N, -6 (30). 31

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 58E: a. Prove that 10n(1)n(mod11) for every positive integer n. b. Prove that a positive integer z is...
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[Number Theory] How do you solve question 2? The second picture is for definitions

A more effective test for quadratic residues is given by Gauss's Lemma: Theorem 7.9 If p is an odd prime and a € Up, then (‡) = (−1)ª where µ = |aP^N\. Before proving this, let us consider an example: Example 7.7 Let p = 19, so P = {1,2,...,9}, and let a = 11. If we multiply each element of P by 11 mod (19), and then represent it by an element of PUN, we get aP = 11P = {-8, 3, -5, 6, -2, 9, 1, -7,4}. This contains four elements of N (the terms with minus signs), so μ = 4, which is even; thus (1) = 1, so 11 € Q19. In fact, 11 = 7² mod (19).
Transcribed Image Text:A more effective test for quadratic residues is given by Gauss's Lemma: Theorem 7.9 If p is an odd prime and a € Up, then (‡) = (−1)ª where µ = |aP^N\. Before proving this, let us consider an example: Example 7.7 Let p = 19, so P = {1,2,...,9}, and let a = 11. If we multiply each element of P by 11 mod (19), and then represent it by an element of PUN, we get aP = 11P = {-8, 3, -5, 6, -2, 9, 1, -7,4}. This contains four elements of N (the terms with minus signs), so μ = 4, which is even; thus (1) = 1, so 11 € Q19. In fact, 11 = 7² mod (19).
1. Evaluate the following Legendre symbols: 2. Use Gauss' Lemma, (i) to compute (ii) (iii) In (ii) p is an odd prime, p ‡ 7; in (iii) p is an odd prime p ‡ 3. In (ii) express the answer in terms of a congruence condition on p mod 28, and in (iii) express the answer in terms of a congruence condition on p mod 24, analogous to = 1 if p = 1 mod 4 and -1 if p = 3 mod 4. 463 541 = (-1)", 4. For an odd prime p, suppose that 31 8933 104729 3. For an odd prime p≥ 7, show that there are consecutive quadratic residues mod p, i.e., that there exists an element a € Up such that μ = |aPN, a (²) - ( ² + ¹) - = Hint: First show that at least one of the numbers 2, 5 and 10 is a quadratic residue mod p. x = a = 1 and consider the congruence = 1. = x² = a mod p. (i) For p = 3 + 4k, i.e., for p = 3 mod 4, show that the solutions to the congruence are tak+1 or x = 1 (ii) For p = 5 + 8k, i.e., for p = 5 mod 8, show that the solutions to the congruence are x = ±ak+¹, +2²k+1 k+1 a Explain how to see which is the correct choice. (iii) Use the results of (i) and (ii) to find the solutions to the congruences x² = 23 mod 751 x² = 15 mod 797
Transcribed Image Text:1. Evaluate the following Legendre symbols: 2. Use Gauss' Lemma, (i) to compute (ii) (iii) In (ii) p is an odd prime, p ‡ 7; in (iii) p is an odd prime p ‡ 3. In (ii) express the answer in terms of a congruence condition on p mod 28, and in (iii) express the answer in terms of a congruence condition on p mod 24, analogous to = 1 if p = 1 mod 4 and -1 if p = 3 mod 4. 463 541 = (-1)", 4. For an odd prime p, suppose that 31 8933 104729 3. For an odd prime p≥ 7, show that there are consecutive quadratic residues mod p, i.e., that there exists an element a € Up such that μ = |aPN, a (²) - ( ² + ¹) - = Hint: First show that at least one of the numbers 2, 5 and 10 is a quadratic residue mod p. x = a = 1 and consider the congruence = 1. = x² = a mod p. (i) For p = 3 + 4k, i.e., for p = 3 mod 4, show that the solutions to the congruence are tak+1 or x = 1 (ii) For p = 5 + 8k, i.e., for p = 5 mod 8, show that the solutions to the congruence are x = ±ak+¹, +2²k+1 k+1 a Explain how to see which is the correct choice. (iii) Use the results of (i) and (ii) to find the solutions to the congruences x² = 23 mod 751 x² = 15 mod 797
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