Question

We consider the following property of a Euclidean function on an integral domain \(R\) : $$ d(a b) \geq d(b) \text { for all } a, b \in R \backslash\\{0\\} \text {. } $$ Our two familiar examples, the degree on \(F[x]\) for a field \(F\) and the absolute value on Z, both fulfill this property. This exercise shows that every Euclidean domain has such a Euclidean function. (i) Show that \(\delta: \mathbb{Z} \longrightarrow \mathbb{N}\) with \(\delta(3)=2\) and \(\delta(a)=|a|\) if \(a \neq 3\) is a Euclidean function on \(\mathbb{Z}\) violating (9). (ii) Suppose that \(R\) is a Euclidean domain and \(D=\\{\delta: \delta\) is a Euclidean function on \(R\\}\). Then \(D\) is nonempty, and we may define a function \(d: R \rightarrow N \cup\\{-\infty\\}\) by \(d(a)=\min \\{\delta(a): \delta \in D\\}\). Show that \(d\) is a Euclidean function on \(R\) (called the minimal Euclidean function). (iii) Let \(\delta\) be a Euclidean function on \(R\) such that \(\delta(a b)<\delta(b)\) for some \(a, b \in R \backslash\\{0\\}\). Find another Euclidean function \(\delta^{*}\) that is smaller than \(\delta\). Conclude that the minimal Euclidean function \(d\) satisfies (9). (iv) Show that for all \(a, b \in R \backslash\\{0\\}\) and a Euclidean function \(d\) satisfying \((9)\), we have \(d(0)

Short answer

Each part proves properties of Euclidean functions and their minimal forms in given scenarios.

Step-by-step solution

Understand the Euclidean Function Concept

In general, a Euclidean function \(\delta\) is a function that assigns a non-negative integer to each non-zero element of a ring. It helps in defining a division algorithm for Euclidean domains.

Solve Part (i)

For \(\delta: \mathbb{Z} \to \mathbb{N}\) with \(\delta(3)=2\) and \(\delta(a)=|a|\) otherwise, we need to show it's a Euclidean function but violates inequality (9). Here, for \(a = 6\) and \(b = 3\), \(\delta(6) = 6\) and \(\delta(3) = 2\), hence \(\delta(6) \geq \delta(3)\) is true. However, this example violates the strict inequality part if \(\delta(ab) < \delta(b)\) for some \(a, b\). Thus, when \(a=3, b=3\), \(\delta(ab)=9 > \delta(b)=2\).

Solve Part (ii)

Given \(R\) is a Euclidean domain, define \(D=\{\delta : \delta \text{ is a Euclidean function on } R\}\). \(D\) is nonempty since \(R\) is Euclidean. Define \(d(a)=\min \{\delta(a): \delta \in D\}\). Since \(D\) is nonempty, \(d(a)\) is well-defined. This results in \(d\) being a Euclidean function on \(R\), and it allows for the Euclidean algorithm in \(R\).

Solve Part (iii)

If \(\delta(a b)<\delta(b)\) for some \(a, b eq 0\), define \(\delta^*(c) = \min(\delta(c), \delta(a b))\). Hence, \(\delta^*(a b) \leq \delta(a b) < \delta(b)\). Minimal Euclidean function \(d\) satisfies (9) as it considers the minimum values of such functions.

Solve Part (iv)

Assuming \(d\) satisfies (9), \(d(0) d(b)\).

Solve Part (v)

With \(d\) as the minimal Euclidean function, \(d(0)=-\infty\) ensures \(0\) is minimal. The group of units \(R^{\times}=\{a \in R: d(a)=0\}\) since any element \(a\) such that \(d(a)=0\) can be viewed as a unit in \(R\).

Solve Part (vi)

For \(F[x]\), consider \(d(a)=\deg a\), which is minimal by polynomial degree definitions. For \(\mathbb{Z}\), the function \(d(a)=|a|\) directly corresponds to the divisor property. Both cases result in \(d(0)=-\infty\) as required by minimality.

Key concepts

Euclidean function

At the heart of a Euclidean domain is the Euclidean function. This function, denoted as \(\delta\), maps from a ring \(R\) to the non-negative integers \(\mathbb{N}\). It's essential in defining what we call a division algorithm, much like how integers work under division. Such a function enables us to determine remainders and quotients, core components of the Euclidean algorithm.

In mathematical terms, the Euclidean function assigns a value to each nonzero element of the ring, aiming to simplify calculations within that ring. One of the key expectations is that it allows for a process similar to the division of integers, where we can achieve a remainder that is smaller than the divisor.

For example, when considering \(\mathbb{Z}\)\ (the integers), the absolute value function \(\delta(a) = |a|\) serves this role. It ensures that when dividing two numbers, we keep the remainders under control. This property is precisely what helps make a quotient and remainder system within the ring possible.

minimal Euclidean function

The minimal Euclidean function is a refined version of a Euclidean function, carefully constructed to offer the smallest values possible while maintaining essential properties. Given any Euclidean domain \(R\), there exists a set \(D\) of all possible Euclidean functions. The minimal Euclidean function, denoted by \(d(a)\), is defined as the smallest possible value for any \(a\) that's still consistent with being a Euclidean function. Essentially, \(d(a) = \min \{ \delta(a) : \delta \in D \}\).

This minimal function excels because it ensures efficiency. By choosing the smallest possible values, it prevents unnecessary complexity and computational effort, which is extremely advantageous when conducting division algorithm operations. It aids in streamlining the Euclidean algorithm since it provides the tightest upper bound for the function values.

A practical example is in the polynomial ring \(F[x]\), where the degree function \(d(a) = \deg(a)\) acts as the minimal Euclidean function. This function not only upholds the needed properties but uses the natural degree of polynomials to ensure the most straightforward calculations.

integral domain

An integral domain is a type of ring with specific properties that make it similar to the set of integers. It is a commutative ring, meaning that the multiplication operation is commutative (i.e., \(a \times b = b \times a\)) and there is a multiplicative identity (an element which acts as a neutral factor when multiplying).

Importantly, an integral domain has no zero divisors. This means that if \(ab = 0\), then either \(a = 0\) or \(b = 0\). This property is crucial because it guarantees that you won't be able to "cancel out" elements to yield zero unless you're starting with zero itself.

The significance of these properties is vast, especially in the application of the Euclidean function. Since the Euclidean domain must also be an integral domain, it ensures that the division algorithm works without unexpected results or undefined operations. This stability and predictability make integral domains a fundamental part of abstract algebra.