Question

Determine the units in $Z[i]$.

Name: Problem 32, Chapter 18: Contemporary Abstract Algebra
Uploaded: 2022-02-07T05:39:23-08:00
Duration: 1 min 28 s
Channel: Manik Pulyani
Description: Determine the units in Z[i].

    Determine the units in $Z[i]$.

Contemporary Abstract Algebra

Joseph Gallian 9th Edition

Chapter 18, Problem 32

Instant Answer

Solved by Expert Manik Pulyani

02/07/2022

Step 1

An element $ u $ in a ring $ R $ is called a unit if there exists an element $ v $ in $ R $ such that $ u \cdot v = 1 $, where $ 1 $ is the multiplicative identity in $ R $. Show more…

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Determine the units in $Z[i]$.

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Key Concepts

Norm Function in Quadratic Integer Rings

The norm function in the context of Gaussian integers is defined by N(a + bi) = a² + b². This function is multiplicative, meaning that the norm of a product equals the product of the norms. It is used to determine invertibility: an element is a unit if and only if its norm is 1, linking the concept of norms directly to the units of the ring.

Gaussian Integers

The Gaussian integers, denoted as Z[i], are complex numbers of the form a + bi where a and b are integers. This set forms a ring and is a central object in algebraic number theory, serving as an extension of the integers into the complex plane.

Units in a Ring

Units in a ring are elements that have a multiplicative inverse within the same ring. In any ring with unity, the collection of all units forms a group under multiplication, and these units play a key role in understanding the factorization properties within the ring.

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