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2. Let R be the subring {a+b√√8i: a, b Z} CC. Define N(a + b√√/8i) = a² + 86². (i) Show that N(wz) = N(w)N(z) for all w, z E R. (ii) Show that when u is a unit of R, N(u) = 1. Hence show that the units of R are 1 and -1. (iii) Show that any element ER with N(x) = 9 is irreducible in R. (iv) By considering (1- √8i)(1√8i), show that R does not have unique factorisation.

2. Let R be the subring {a+b√√8i: a, b Z} CC. Define N(a + b√√/8i) = a² + 86². (i) Show that N(wz) = N(w)N(z) for all w, z E R. (ii) Show that when u is a unit of R, N(u) = 1. Hence show that the units of R are 1 and -1. (iii) Show that any element ER with N(x) = 9 is irreducible in R. (iv) By considering (1- √8i)(1√8i), show that R does not have unique factorisation.

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter3: Functions And Graphs
Section3.2: Graphs Of Equations
Problem 67E
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2. Let R be the subring {a+b√√8i: a, b Z} CC. Define N(a + b√√/8i) = a² + 86². (i) Show that N(wz) = N(w)N(z) for all w, z E R. (ii) Show that when u is a unit of R, N(u) = 1. Hence show that the units of R are 1 and -1. (iii) Show that any element ER with N(x) = 9 is irreducible in R. (iv) By considering (1- √8i)(1√8i), show that R does not have unique factorisation.
Transcribed Image Text:2. Let R be the subring {a+b√√8i: a, b Z} CC. Define N(a + b√√/8i) = a² + 86². (i) Show that N(wz) = N(w)N(z) for all w, z E R. (ii) Show that when u is a unit of R, N(u) = 1. Hence show that the units of R are 1 and -1. (iii) Show that any element ER with N(x) = 9 is irreducible in R. (iv) By considering (1- √8i)(1√8i), show that R does not have unique factorisation.
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