Consider the integral domain ℤ[(√−2 )] = {a + i b√2 : a, b ∈ ℤ} and define N on ℤ[(√−2 )] by N(a + i b√2 ) = a2 + 2b2. Prove that (a) N is a multiplicative norm on ℤ(√−2 ), and (b) α ∈ ℤ[√−2 ] is a unit if and only if N(α) = 1.

Consider the integral domain ℤ[(√−2 )] = {a + i b√2 : a, b ∈ ℤ} and define N on ℤ[(√−2 )] by N(a + i b√2 ) = a2 + 2b2. Prove that (a) N is a multiplicative norm on ℤ(√−2 ), and (b) α ∈ ℤ[√−2 ] is a unit if and only if N(α) = 1.

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter6: More On Rings

Section6.3: The Characteristic Of A Ring

Problem 9E: Let D be an integral domain with four elements, D=0,e,a,b, where e is the unity. a. Prove that D has...

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