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Exercise 4. Recall that Z(i) denote the Gaussian integers Z(i) = {a + bi | a, b = Z} 1. Show that Z(i) is a Euclidean Domain with the norm given by the square of complex norm. N(a+bi) = a² +6² 2. Show that the norm is multiplicative, meaning that for 2₁, 22 € Z(i), N(2122) = N(2₁)N(2₂)

Exercise 4. Recall that Z(i) denote the Gaussian integers Z(i) = {a + bi | a, b = Z} 1. Show that Z(i) is a Euclidean Domain with the norm given by the square of complex norm. N(a+bi) = a² +6² 2. Show that the norm is multiplicative, meaning that for 2₁, 22 € Z(i), N(2122) = N(2₁)N(2₂)

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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Theorem 1. If p is a prime number and p = 1 mod 4, then p|m² + 1 for some m € Z. Exercise 4. Recall that Z(i) denote the Gaussian integers Z(i) = {a + bi | a,b ≤ Z} 1. Show that Z(i) is a Euclidean Domain with the norm given by the square of complex norm. N(a+bi) = a² +6² 2. Show that the norm is multiplicative, meaning that for 21, 22 € Z(i), N(2₁22) = N(2₁)N(22) 3. Use the above theorem to prove that if p E N is a prime number such that p = 1 mod 4, then p is reducible in Z(i). 4. Prove that if n E N has a prime factorization α2 n = 2kp¹p2² an B1 B2 Pn 91 92 Bm •9,8m where p's are distinct prime numbers congruent to 1 mod 4; q;'s are distinct prime numbers congruent to -1 mod 4; and 3;'s are even number, then n can be written as a product of two primes.
Transcribed Image Text:Theorem 1. If p is a prime number and p = 1 mod 4, then p|m² + 1 for some m € Z. Exercise 4. Recall that Z(i) denote the Gaussian integers Z(i) = {a + bi | a,b ≤ Z} 1. Show that Z(i) is a Euclidean Domain with the norm given by the square of complex norm. N(a+bi) = a² +6² 2. Show that the norm is multiplicative, meaning that for 21, 22 € Z(i), N(2₁22) = N(2₁)N(22) 3. Use the above theorem to prove that if p E N is a prime number such that p = 1 mod 4, then p is reducible in Z(i). 4. Prove that if n E N has a prime factorization α2 n = 2kp¹p2² an B1 B2 Pn 91 92 Bm •9,8m where p's are distinct prime numbers congruent to 1 mod 4; q;'s are distinct prime numbers congruent to -1 mod 4; and 3;'s are even number, then n can be written as a product of two primes.
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