that N(ry) = N(x)N(y) for all x and y in 22). 2. If x is invertible in Z[1], show that N(x) must equal 1. 3. Conclude that the only units of Z are ±1 and ±.
that N(ry) = N(x)N(y) for all x and y in 22). 2. If x is invertible in Z[1], show that N(x) must equal 1. 3. Conclude that the only units of Z are ±1 and ±.
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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Question
abstract algebra
![**Exercise 2.113**
This exercise determines the units of the ring \( \mathbb{Z}[i] \):
1. Define a function \( N : \mathbb{Z}[i] \to \mathbb{Z} \) by \( N(a + bi) = a^2 + b^2 \). Show
(There are no graphs or diagrams in this image.)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3d940ce8-cba2-4a95-af25-aae0739ca5aa%2Fe48a4f36-423e-4f18-998d-9e1dcb6dcecd%2Fbpjeaf_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Exercise 2.113**
This exercise determines the units of the ring \( \mathbb{Z}[i] \):
1. Define a function \( N : \mathbb{Z}[i] \to \mathbb{Z} \) by \( N(a + bi) = a^2 + b^2 \). Show
(There are no graphs or diagrams in this image.)
![1. Prove that \( N(xy) = N(x)N(y) \) for all \( x \) and \( y \) in \( \mathbb{Z}[i] \).
2. If \( x \) is invertible in \( \mathbb{Z}[i] \), show that \( N(x) \) must equal 1.
3. Conclude that the only units of \( \mathbb{Z}[i] \) are \(\pm 1\) and \(\pm i\).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3d940ce8-cba2-4a95-af25-aae0739ca5aa%2Fe48a4f36-423e-4f18-998d-9e1dcb6dcecd%2F15qzi4_processed.jpeg&w=3840&q=75)
Transcribed Image Text:1. Prove that \( N(xy) = N(x)N(y) \) for all \( x \) and \( y \) in \( \mathbb{Z}[i] \).
2. If \( x \) is invertible in \( \mathbb{Z}[i] \), show that \( N(x) \) must equal 1.
3. Conclude that the only units of \( \mathbb{Z}[i] \) are \(\pm 1\) and \(\pm i\).
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