Prove that, for mE Z and 2 = a +b√a if and only if m and m | b in Z. (b) Prove that, if 2₁, 22 € Z[√d] and 2₁ | 22 in Z[√d], then Na(21) | Na(22) in Z. m 2 in
Prove that, for mE Z and 2 = a +b√a if and only if m and m | b in Z. (b) Prove that, if 2₁, 22 € Z[√d] and 2₁ | 22 in Z[√d], then Na(21) | Na(22) in Z. m 2 in
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 5: Mathematical Proofs Involving Number Theory**
Let \( d \in \mathbb{Z} - \{0\} \) and assume \( \sqrt{d} \notin \mathbb{Q} \).
1. **Part (a)**: Prove that, for \( m \in \mathbb{Z} \) and \( z = a + b\sqrt{d} \in \mathbb{Z}[\sqrt{d}] \), \( m \mid z \) in \( \mathbb{Z}[\sqrt{d}] \) if and only if \( m \mid a \) and \( m \mid b \) in \( \mathbb{Z} \).
2. **Part (b)**: Prove that, if \( z_1, z_2 \in \mathbb{Z}[\sqrt{d}] \) and \( z_1 \mid z_2 \) in \( \mathbb{Z}[\sqrt{d}] \), then \( N_d(z_1) \mid N_d(z_2) \) in \( \mathbb{Z} \).
**Hint**: This problem relates to definitions and previously established facts about \( \mathbb{Z}[\sqrt{d}] \) and its norm function.
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**Explanation**:
- **Part (a)** involves showing an equivalence relationship regarding divisibility within the ring \( \mathbb{Z}[\sqrt{d}] \) and in the integers \( \mathbb{Z} \).
- **Part (b)** involves proving a property of the norm function \( N_d \) and how it interacts with divisibility within \( \mathbb{Z}[\sqrt{d}] \).
Understanding and leveraging the properties and definitions of the integers, the ring \( \mathbb{Z}[\sqrt{d}] \), and the norm function within this ring are crucial for successfully tackling these proofs.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Febb913e1-4986-4d74-b6ce-ad576ddf43d3%2F84e1baf5-4c36-4574-9e11-295d43dfd6de%2Fdtecql5_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem 5: Mathematical Proofs Involving Number Theory**
Let \( d \in \mathbb{Z} - \{0\} \) and assume \( \sqrt{d} \notin \mathbb{Q} \).
1. **Part (a)**: Prove that, for \( m \in \mathbb{Z} \) and \( z = a + b\sqrt{d} \in \mathbb{Z}[\sqrt{d}] \), \( m \mid z \) in \( \mathbb{Z}[\sqrt{d}] \) if and only if \( m \mid a \) and \( m \mid b \) in \( \mathbb{Z} \).
2. **Part (b)**: Prove that, if \( z_1, z_2 \in \mathbb{Z}[\sqrt{d}] \) and \( z_1 \mid z_2 \) in \( \mathbb{Z}[\sqrt{d}] \), then \( N_d(z_1) \mid N_d(z_2) \) in \( \mathbb{Z} \).
**Hint**: This problem relates to definitions and previously established facts about \( \mathbb{Z}[\sqrt{d}] \) and its norm function.
---
**Explanation**:
- **Part (a)** involves showing an equivalence relationship regarding divisibility within the ring \( \mathbb{Z}[\sqrt{d}] \) and in the integers \( \mathbb{Z} \).
- **Part (b)** involves proving a property of the norm function \( N_d \) and how it interacts with divisibility within \( \mathbb{Z}[\sqrt{d}] \).
Understanding and leveraging the properties and definitions of the integers, the ring \( \mathbb{Z}[\sqrt{d}] \), and the norm function within this ring are crucial for successfully tackling these proofs.
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Can further explanation please be provided for part a? I am having trouble understanding how we are supposed to equeate the real and imaginary parts on both sides, and everything after that step
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