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.
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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![Consider the quotient ring R1 = Z3[x]/(x² + 1) and R2 = Z3[i] = {a + bi / a, b E Z3} with operations
(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 : Rị → R2 between these rings. (Explain the operations in R1 and how
ø respects them).
(b) Show that x2 +1 is irreducible in Z3[]. So R1 - R2 is a field.
(c) Find (1+ 2i)-1 in this field.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F7fd1602c-4cb7-4ab7-b037-059fcc7920df%2F2bd2b68c-6d86-4d43-8c01-21406e1fea02%2Fcbm73nh_processed.png&w=3840&q=75)
Transcribed Image Text:Consider the quotient ring R1 = Z3[x]/(x² + 1) and R2 = Z3[i] = {a + bi / a, b E Z3} with operations
(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 : Rị → R2 between these rings. (Explain the operations in R1 and how
ø respects them).
(b) Show that x2 +1 is irreducible in Z3[]. So R1 - R2 is a field.
(c) Find (1+ 2i)-1 in this field.
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