Use the norm map N: Z [√-3] → Z2o by N(a+b√-3) = a² + 36² (you can assume it is multiplicative, i.e N(wz) = N(w)N(z)) to prove (a) The only units of Z [√-3] are +1 (b) Z[√-3] is not a UFD in two ways: (i) by producing an irreducible element that isn't prime and showing the uniqueness into irreducibles is violated in this ring (like we did for Z [√5] )
Use the norm map N: Z [√-3] → Z2o by N(a+b√-3) = a² + 36² (you can assume it is multiplicative, i.e N(wz) = N(w)N(z)) to prove (a) The only units of Z [√-3] are +1 (b) Z[√-3] is not a UFD in two ways: (i) by producing an irreducible element that isn't prime and showing the uniqueness into irreducibles is violated in this ring (like we did for Z [√5] )
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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![**Title: Exploring the Norm Map and Unique Factorization Domain (UFD) in Integer Rings**
**Introduction:**
In this lesson, we will explore the concept of the norm map in the integer ring extended by the square root of -3, denoted as \( \mathbb{Z}[\sqrt{-3}] \). We will use the norm map to prove certain properties about units and factorization within this ring.
**Norm Map:**
The norm map is a function defined as \( N: \mathbb{Z}[\sqrt{-3}] \to \mathbb{Z}_{\geq 0} \), which is given by:
\[
N(a + b\sqrt{-3}) = a^2 + 3b^2
\]
This map is multiplicative, meaning that for any \( w, z \in \mathbb{Z}[\sqrt{-3}] \), we have:
\[
N(wz) = N(w)N(z)
\]
**Proof Objectives:**
**(a) Units in \( \mathbb{Z}[\sqrt{-3}] \):**
We aim to show that the only units in \( \mathbb{Z}[\sqrt{-3}] \) are \( \pm 1 \). A unit is an element with a multiplicative inverse also in the ring. By using properties of the norm function, we can examine \( N(u) = 1 \) for any unit \( u \), leading us to conclude that the possible units are \( \pm 1 \).
**(b) Non-UFD Property of \( \mathbb{Z}[\sqrt{-3}] \):**
A Unique Factorization Domain (UFD) is a ring in which every element can be factored uniquely into irreducible elements, up to units. To prove that \( \mathbb{Z}[\sqrt{-3}] \) is not a UFD, we will:
1. Produce an irreducible element that is not prime.
2. Show that the uniqueness of factorization into irreducibles is violated in this ring.
This approach is similar to what is done for \( \mathbb{Z}[\sqrt{-5}] \), where specific examples of non-unique factorizations can be illustrated.
**Conclusion:**
Understanding the properties of units and the factorization structure in \( \mathbb{Z}[\sqrt{-3}] \) using](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9c55fd55-ae67-4b97-a36c-91359ff73a6f%2F0490efbd-a007-420b-ab26-2b00e3bf9919%2F0khvesf_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Title: Exploring the Norm Map and Unique Factorization Domain (UFD) in Integer Rings**
**Introduction:**
In this lesson, we will explore the concept of the norm map in the integer ring extended by the square root of -3, denoted as \( \mathbb{Z}[\sqrt{-3}] \). We will use the norm map to prove certain properties about units and factorization within this ring.
**Norm Map:**
The norm map is a function defined as \( N: \mathbb{Z}[\sqrt{-3}] \to \mathbb{Z}_{\geq 0} \), which is given by:
\[
N(a + b\sqrt{-3}) = a^2 + 3b^2
\]
This map is multiplicative, meaning that for any \( w, z \in \mathbb{Z}[\sqrt{-3}] \), we have:
\[
N(wz) = N(w)N(z)
\]
**Proof Objectives:**
**(a) Units in \( \mathbb{Z}[\sqrt{-3}] \):**
We aim to show that the only units in \( \mathbb{Z}[\sqrt{-3}] \) are \( \pm 1 \). A unit is an element with a multiplicative inverse also in the ring. By using properties of the norm function, we can examine \( N(u) = 1 \) for any unit \( u \), leading us to conclude that the possible units are \( \pm 1 \).
**(b) Non-UFD Property of \( \mathbb{Z}[\sqrt{-3}] \):**
A Unique Factorization Domain (UFD) is a ring in which every element can be factored uniquely into irreducible elements, up to units. To prove that \( \mathbb{Z}[\sqrt{-3}] \) is not a UFD, we will:
1. Produce an irreducible element that is not prime.
2. Show that the uniqueness of factorization into irreducibles is violated in this ring.
This approach is similar to what is done for \( \mathbb{Z}[\sqrt{-5}] \), where specific examples of non-unique factorizations can be illustrated.
**Conclusion:**
Understanding the properties of units and the factorization structure in \( \mathbb{Z}[\sqrt{-3}] \) using
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