This question is about the division algorithm in Z[√2]. Let w = (-3+8√2) and 2 = (6-7√2). (0) Find the quotient q and remainder r when w is divided by z in Z[√2]. Find the quotient 9₁ and remainder r₁ when z is divided by r above. Show that is a unit in Z[√2] and explain how this implies that w and z are coprime in in Z[√2]. Hence find the inverse of z in the quotient ring Z[√2]/wZ[√2].
This question is about the division algorithm in Z[√2]. Let w = (-3+8√2) and 2 = (6-7√2). (0) Find the quotient q and remainder r when w is divided by z in Z[√2]. Find the quotient 9₁ and remainder r₁ when z is divided by r above. Show that is a unit in Z[√2] and explain how this implies that w and z are coprime in in Z[√2]. Hence find the inverse of z in the quotient ring Z[√2]/wZ[√2].
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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![This question is about the division algorithm in Z[√2]. Let
w = (-3+8√2)
and
= (6-7√2).
Find the quotient q and remainder r when w is divided by z in Z[√2].
(ii) (
Find the quotient 9₁ and remainder r₁ when z is divided by r above. Show that
r₁ is a unit in Z[√2] and explain how this implies that w and z are coprime in in
Z[√2].
(iii)
Hence find the inverse of z in the quotient ring
Z[√2]/wZ[√2].](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8d2926bd-da4f-4f1e-96bb-938076db5805%2F37a8d9c9-9798-4261-983e-ad09f1527bb8%2Fjq0piw.jpeg&w=3840&q=75)
Transcribed Image Text:This question is about the division algorithm in Z[√2]. Let
w = (-3+8√2)
and
= (6-7√2).
Find the quotient q and remainder r when w is divided by z in Z[√2].
(ii) (
Find the quotient 9₁ and remainder r₁ when z is divided by r above. Show that
r₁ is a unit in Z[√2] and explain how this implies that w and z are coprime in in
Z[√2].
(iii)
Hence find the inverse of z in the quotient ring
Z[√2]/wZ[√2].
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