Consider the groups Z[i] = {a + bi | a, b = Z, i² = -1} and Z (both under +), and the function6: Z[i] → Z given by o(a + bi) = a.(a) Show that satisfies the homomorphism property.(b) Is an isomorphism? Justify your answer.

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Answered: Consider the groups Z[i] = {a + bi | a,…

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter3: Groups

Section3.6: Homomorphisms

Problem 3E: 3. Consider the additive groups of real numbers and complex numbers and define by . Prove that is...

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Consider the groups Z[i] = {a + bi | a, b = Z, i² = -1} and Z (both under +), and the function o: Z[i] → Z given by o(a + bi) = a. (a) Show that satisfies the homomorphism property. (b) Is an isomorphism? Justify your answer.