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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].

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter4: Polynomial And Rational Functions
Section4.2: Properties Of Division
Problem 26E
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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].
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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