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3. Let p be an odd prime. Define sets A and B by A = {a € Z: 1 ≤ a ≤ (p-1)/2} and B = {b Z: (p+1)/2 ≤ b ≤p-1}. (a) Give A and B in terms of p.² (b) Briefly explain why each element of Fx has a unique representative in AUB. (c) Prove using basic facts about inequalities that if a € A, then p- a € B. (d) By (c), the function f: A → B given by f(a) = p-a is well-defined. Prove that f is bijective. (Hint: Show that f is injective and then invoke the following fact: if S and T are finite sets with |S| = |T and F: S→T is injective, then F is bijective. Or you could exhibit an inverse for f.) (e) Prove that (a+pZ)² = (f(a) +pZ)². (f) Prove that the number of squares in FX is (p-1)/2, i.e., exactly half of the elements in F are squares. (Hint: Show that the function g: A → {a²: a € F} given by g(a) = (a +pZ)2 is bijective. Parts (b) and (e) will help with surjectivity. For injectivity, consider a₁, a2 E A with g(a₁) = g(az). Deduce as a consequence that a₁ +pZ = ±a₂+pZ. Rule out the equality a₁ +pZ = -a₂+pZ, and then use (b) to prove that a₁ = a₂.)

3. Let p be an odd prime. Define sets A and B by A = {a € Z: 1 ≤ a ≤ (p-1)/2} and B = {b Z: (p+1)/2 ≤ b ≤p-1}. (a) Give A and B in terms of p.² (b) Briefly explain why each element of Fx has a unique representative in AUB. (c) Prove using basic facts about inequalities that if a € A, then p- a € B. (d) By (c), the function f: A → B given by f(a) = p-a is well-defined. Prove that f is bijective. (Hint: Show that f is injective and then invoke the following fact: if S and T are finite sets with |S| = |T and F: S→T is injective, then F is bijective. Or you could exhibit an inverse for f.) (e) Prove that (a+pZ)² = (f(a) +pZ)². (f) Prove that the number of squares in FX is (p-1)/2, i.e., exactly half of the elements in F are squares. (Hint: Show that the function g: A → {a²: a € F} given by g(a) = (a +pZ)2 is bijective. Parts (b) and (e) will help with surjectivity. For injectivity, consider a₁, a2 E A with g(a₁) = g(az). Deduce as a consequence that a₁ +pZ = ±a₂+pZ. Rule out the equality a₁ +pZ = -a₂+pZ, and then use (b) to prove that a₁ = a₂.)

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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3. Let p be an odd prime. Define sets A and B by A = {a € Z : 1 ≤ a ≤ (p-1)/2} and B = {b € Z: (p+1)/2 ≤ b ≤p-1}. (a) Give A and B in terms of p.² (b) Briefly explain why each element of Fx has a unique representative in AUB. (c) Prove using basic facts about inequalities that if a € A, then p- a € B. (d) By (c), the function f : A → B given by f(a) = p-a is well-defined. Prove that f is bijective. (Hint: Show that f is injective and then invoke the following fact: if S and T are finite sets with |S| = |T| and F: S→T is injective, then F is bijective. Or you could exhibit an inverse for f.) (e) Prove that (a +pZ)² = (ƒ(a) +pZ)². (f) Prove that the number of squares in F is (p-1)/2, i.e., exactly half of the elements in F are squares. (Hint: Show that the function g: A → {a² a € Fx} given by g(a) = (a +pZ)2 is bijective. Parts (b) and (e) will help with surjectivity. For injectivity, consider a₁,02 € A with g(a₁) = g(a₂). Deduce as a consequence that a₁ +pZ = ±a₂+pZ. Rule out the equality a₁ +pZ = -a₂+pZ, and then use (b) to prove that a₁ = a₂.)
Transcribed Image Text:3. Let p be an odd prime. Define sets A and B by A = {a € Z : 1 ≤ a ≤ (p-1)/2} and B = {b € Z: (p+1)/2 ≤ b ≤p-1}. (a) Give A and B in terms of p.² (b) Briefly explain why each element of Fx has a unique representative in AUB. (c) Prove using basic facts about inequalities that if a € A, then p- a € B. (d) By (c), the function f : A → B given by f(a) = p-a is well-defined. Prove that f is bijective. (Hint: Show that f is injective and then invoke the following fact: if S and T are finite sets with |S| = |T| and F: S→T is injective, then F is bijective. Or you could exhibit an inverse for f.) (e) Prove that (a +pZ)² = (ƒ(a) +pZ)². (f) Prove that the number of squares in F is (p-1)/2, i.e., exactly half of the elements in F are squares. (Hint: Show that the function g: A → {a² a € Fx} given by g(a) = (a +pZ)2 is bijective. Parts (b) and (e) will help with surjectivity. For injectivity, consider a₁,02 € A with g(a₁) = g(a₂). Deduce as a consequence that a₁ +pZ = ±a₂+pZ. Rule out the equality a₁ +pZ = -a₂+pZ, and then use (b) to prove that a₁ = a₂.)
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