Problem 15 - A First Course in Abstract Algebra

Question

Let \(D\) be a Euclidean domain and let \(v\) be a Euclidean norm on \(D\). Show that if \(a\) and \(b\) are associates in \(D\). then \(v(a)=v(b)\).

Step-by-Step Solution

Answer

If \(a\) and \(b\) are associates, then \(v(a) = v(b)\) because multiplication by a unit does not change the Euclidean norm.

Step 1: Define the Concept of Associates

In a Euclidean domain, two elements \(a\) and \(b\) are called associates if there exists a unit \(u\) such that \(b = ua\). This means that \(a\) and \(b\) are essentially the same up to multiplication by a unit, which is an invertible element in the domain.

Step 2: Recognize Properties of Euclidean Norm

A Euclidean norm \(v\) is a function from the non-zero elements of \(D\) to the non-negative integers such that for any non-zero \(a, b \in D\), if \(b eq 0\), there are uniquely determined \(q, r \in D\) with \(a = bq + r\) where either \(r=0\) or \(v(r) < v(b)\). The norm measures size or magnitude, preserving properties under multiplication by units.

Step 3: Express Norms of Associates

Given that \(b = ua\) where \(u\) is a unit, we investigate the norm. Notice that \(v(ua) = v(u)v(a)\) by definition of norm in Euclidean domain where \(v(u) = 1\) since \(u\) is a unit.

Step 4: Prove Norm Equality

Because a unit \(u\) in a Euclidean domain has a norm \(v(u) = 1\), the norm of \(b = ua\) is \(v(b) = v(ua) = v(u) \cdot v(a) = 1 \cdot v(a) = v(a)\). Thus, the norms of \(a\) and \(b\), \(v(a) = v(b)\), are equal.

Key Concepts

Euclidean norm

In the context of a Euclidean domain, a Euclidean norm is a crucial concept for understanding the structure and behavior of the domain's elements. A Euclidean norm is a function denoted by \( v \) which assigns a non-negative integer to each non-zero element of the domain \( D \). The primary aim of this norm is to measure the 'size' or 'magnitude' of elements within the domain. For any non-zero elements \( a \) and \( b \) in \( D \), if \( b eq 0 \), there exist unique elements \( q \) and \( r \) in \( D \) such that \( a = bq + r \), where either \( r = 0 \) or \( v(r) < v(b) \).
This property ensures that the Euclidean norm is effective in handling division operations within the domain, allowing us to compare sizes and manage remainders in calculations efficiently. It also preserves specific characteristics when elements are multiplied by units, a feature that is essential in solving problems in Euclidean domains.

Associates

Associates are elements within a Euclidean domain that showcase a special relationship: they are essentially the same element up to multiplication by a unit. If you have two elements \( a \) and \( b \) in a Euclidean domain \( D \), they are considered associates if there exists a unit \( u \) such that \( b = ua \). In simpler terms, associates are like reflections of the same number, altered only by a factor that does not affect the core value, thanks to the unit being invertible.
This idea of associates helps in identifying when two elements are just scaled versions of each other without any fundamental change in numerical structure. This factorization plays a critical role in working with elements in a Euclidean domain, especially when comparing magnitudes or calculating norms.

Units

Units in a Euclidean domain are the building blocks that allow for flexibility and reversibility in calculations. These are elements that have a multiplicative inverse within the domain. An element \( u \) in \( D \) is a unit if there exists another element \( u^{-1} \) in \( D \) such that \( uu^{-1} = 1 \).
This means units are invertible, enabling us to manipulate other elements without altering their intrinsic values or properties. In the context of associates, units become essential as they allow us to express one element as a multiplication of another by a unit, implying that they retain the same norm. Understanding units is key for proving various theorems and lemmas within Euclidean domains, especially those involving norm preservation.

Norm equality

Norm equality is a fundamental concept in the study of Euclidean domains. It involves showing that certain elements have identical Euclidean norms, which is crucial for many mathematical proofs and exercises. When dealing with associates \( a \) and \( b \) in a Euclidean domain \( D \), if \( b = ua \) where \( u \) is a unit, norm equality asserts that \( v(a) = v(b) \). This sustaining property comes from the fact that the Euclidean norm is preserved under multiplication by units, because \( v(u) = 1 \).
Proving norm equality involves confirming that multiplying an element by a unit does not change its norm, a concept which plays a pivotal role in many theoretical and computational aspects within Euclidean domains. Recognizing norm equality helps in visualizing elements as fundamentally similar in size when viewed through the lens of Euclidean norms, despite potential superficial differences introduced by units.