3. Let i C be a square root of -1. Use part (c) of the preceding problem to prove that Z[i] = {±1, +i}.
3. Let i C be a square root of -1. Use part (c) of the preceding problem to prove that Z[i] = {±1, +i}.
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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![### Problem 3:
**Statement:**
Let \( i \in \mathbb{C} \) be a square root of \(-1\). Use part (c) of the preceding problem to prove that
\[ \mathbb{Z}[i]^{\times} = \{\pm 1, \pm i\}. \]
**Detailed Explanation:**
- \( i \) is defined as a square root of \(-1\) in the set of complex numbers (\(\mathbb{C}\)).
- You are required to utilize part (c) of the preceding problem (not provided here) to demonstrate that the group of units (invertible elements) in the Gaussian integers (\(\mathbb{Z}[i]\)) equals \(\{\pm 1, \pm i\}\).
**Additional Notes:**
- The Gaussian integers are of the form \(a + bi\) where \(a, b \in \mathbb{Z}\).
- The units in any ring \( R \) are the elements that have a multiplicative inverse in \( R \).
For further in-depth explanations and proofs, please refer to Gaussian integers and units in abstract algebra textbooks or resources.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Febb913e1-4986-4d74-b6ce-ad576ddf43d3%2F39593307-8fc3-4f66-abc1-194d60e29aa9%2F4q35b9q_processed.png&w=3840&q=75)
Transcribed Image Text:### Problem 3:
**Statement:**
Let \( i \in \mathbb{C} \) be a square root of \(-1\). Use part (c) of the preceding problem to prove that
\[ \mathbb{Z}[i]^{\times} = \{\pm 1, \pm i\}. \]
**Detailed Explanation:**
- \( i \) is defined as a square root of \(-1\) in the set of complex numbers (\(\mathbb{C}\)).
- You are required to utilize part (c) of the preceding problem (not provided here) to demonstrate that the group of units (invertible elements) in the Gaussian integers (\(\mathbb{Z}[i]\)) equals \(\{\pm 1, \pm i\}\).
**Additional Notes:**
- The Gaussian integers are of the form \(a + bi\) where \(a, b \in \mathbb{Z}\).
- The units in any ring \( R \) are the elements that have a multiplicative inverse in \( R \).
For further in-depth explanations and proofs, please refer to Gaussian integers and units in abstract algebra textbooks or resources.
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