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 howø 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.

Consider the quotient ring R1=Z3x/x2+1 and R2=Z3i=a+bi/a,b∈Z3 with operations…

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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Question

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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