Consider the subset Z[i] = {a + bi | a, b € Z} of the complex numbers, C. Note: Z[i] is called the set of Gaussian integers. a. Prove that Z[i] is a subring of C. b. Is Z[i] a commutative ring? Justify your answer. c. Is Z[i] a ring with identity? Justify your answer. d. Is Z[i] an integral domain? Prove or disprove. [Hint: C is an integral domain.] e. Is Z[i] a field? Prove or disprove. [Hint: The inverse of the complex number a + bi is 1 a-bi = :] a+bi a²+b²
Consider the subset Z[i] = {a + bi | a, b € Z} of the complex numbers, C. Note: Z[i] is called the set of Gaussian integers. a. Prove that Z[i] is a subring of C. b. Is Z[i] a commutative ring? Justify your answer. c. Is Z[i] a ring with identity? Justify your answer. d. Is Z[i] an integral domain? Prove or disprove. [Hint: C is an integral domain.] e. Is Z[i] a field? Prove or disprove. [Hint: The inverse of the complex number a + bi is 1 a-bi = :] a+bi a²+b²
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter5: Rings, Integral Domains, And Fields
Section5.1: Definition Of A Ring
Problem 32E: 32. Consider the set .
a. Construct addition and multiplication tables for, using the...
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