Consider the quotient ring R1 = Z3[a]/(x² + 1) and R2 = Z3[i] = {a + bi / a,b e Z3} with operations %3D %3D (a + bi) + (a' + b'i) = (a + a') + (b + b')i, (a + bi) · (a' + b'i) = (aa' – bb') + (ab' + a'b)i. (a) Show an isomorphism o : R1 → R2 between these rings. (Explain the operations in Rị and how ø respects them). (b) Show that x² +1 is irreducible in Z3[x]. So R1 = R2 is a field. (c) Find (1+ 2i)-1 in this field.
Consider the quotient ring R1 = Z3[a]/(x² + 1) and R2 = Z3[i] = {a + bi / a,b e Z3} with operations %3D %3D (a + bi) + (a' + b'i) = (a + a') + (b + b')i, (a + bi) · (a' + b'i) = (aa' – bb') + (ab' + a'b)i. (a) Show an isomorphism o : R1 → R2 between these rings. (Explain the operations in Rị and how ø respects them). (b) Show that x² +1 is irreducible in Z3[x]. So R1 = R2 is a field. (c) Find (1+ 2i)-1 in this field.
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter8: Polynomials
Section8.1: Polynomials Over A Ring
Problem 17E
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