Question

Let \(R\) be a Euclidean domain, with a Euclidean function \(d: R \rightarrow N \cup\\{-\infty\\}\) that has the additional properties o \(d(a b)=d(a)+d(b) .\) o \(d(a+b) \leq \max \\{d(a), d(b)\\}\), with equality if \(d(a) \neq d(b)\). o \(d\) is surjective. for all \(a, b \in R\). Prove that \(R\) is a polynomial ring with \(d\) as degree function. Proceed as follows: (i) Prove that \(d(a)=-\infty\) if and only if \(a=0\). (ii) Show that \(F=\\{a \in R: d(a) \leq 0\\}\) is a subfield of \(R\). (iii) Let \(x \in R\) be such that \(d(x)=1\), and prove that every nonzero \(a \in R\) has a unique representation $$ a=a_{n} x^{n}+a_{n-1} x^{n-1}+\cdots+a_{1} x+a_{0} $$ where \(n=d(a), a_{0} \ldots, a_{n} \in F_{\text {, and }} a_{n} \neq 0\).

Short answer

1. \(d(a) = -\infty\) iff \(a = 0\). 2. \(F\) is a subfield. 3. Unique polynomial form in \(x\).

Step-by-step solution

Prove that \(d(a) = -\infty\) if and only if \(a = 0\)

Let's prove the 'if' direction first. If \(a = 0\), then by the definition of a Euclidean domain, the Euclidean function should map zero to the smallest possible value, \(-\infty\), since no positive element divides zero. For the 'only if' direction, assume \(d(a) = -\infty\). If there existed some nonzero element \(b\) such that \(d(b) = -\infty\), then \(d(a + b)\) would by assumption not exceed \(\max\{d(a), d(b)\} = -\infty\). This would imply \(d(a + b) = -\infty\), contradicting the assumption that nonzero elements cannot have \(-\infty\) value for \(d\). Hence, \(a\) must be zero. Therefore, \(d(a) = -\infty\) if and only if \(a = 0\).

Show \(F = \{a \in R: d(a) \leq 0\}\) is a subfield of \(R\)

To show that \(F\) is a subfield, we must prove it is closed under addition, multiplication, and taking inverses.**Addition and Multiplication**: Take any \(a, b \in F\), so \(d(a), d(b) \leq 0\). From the property \(d(a+b)\leq\max\{d(a),d(b)\}\) and \(d(ab) = d(a) + d(b)\), it follows that \(d(a+b)\leq 0\) and \(d(ab) \leq 0\). Hence, both operations preserve membership in \(F\).**Inverses**: For the existence of inverses (excluding zero), for any \(a eq 0\) where \(d(a) \leq 0\), a required consequence of \(d\) being surjective and property of Euclidean domains is that there exists a \(b\) such that \(ab = 1\), making \(d(b) = -d(a)\). Since \(d(b)\) is a non-positive integer, \(b\) also belongs to \(F\). Thus, \(F\) is a subfield.

Represent nonzero \(a \in R\) in terms of \(x\) with \(d(x) = 1\)

Given \(x \in R\) with \(d(x) = 1\), we aim to express each nonzero \(a\) with degrees. Assume \(a eq 0\) with \(d(a) = n\), and set \(a=a_n x^n + a_{n-1} x^{n-1} + \cdots + a_0\). To show uniqueness:1. Assume two alternate expressions for \(a\): \[a = a_n x^n + a_{n-1}x^{n-1} + \ldots + a_0 = b_mx^m + b_{m-1}x^{m-1} + \ldots + b_0 \]2. If \(n eq m\), assume without loss of generality \(n > m\). Subtract the equations: \[a_n x^n + a_{n-1}x^{n-1} + \ldots + a_{m+1}x^{m+1} = (b_0 - a_0) \] The left side has degree \( > m \) while the right side doesn't, contradiction unless all terms \(a_n, a_{n-1}, \ldots\) are zero, forcing equality in expressions. Thus, \(n = m\) and each coefficient must equal, providing uniqueness.Therefore, \(a\) can be uniquely expressed in terms of powers of \(x\) with coefficients in \(F\).

Key concepts

Euclidean Function

When discussing a Euclidean domain, a critical concept is the Euclidean function. This function, denoted as \(d: R \rightarrow \mathbb{N} \cup \{-\infty\}\), is used to measure 'size' in a way that helps to perform Euclidean algorithms within the domain. In simple terms, the Euclidean function assigns non-negative integers to each element of the domain, except zero, which is mapped to \(-\infty\).
The concept of size in this context often correlates with divisibility. For instance, in integer rings, the Euclidean function is the absolute value, while in polynomial rings, it is the degree of the polynomial.
Euclidean functions satisfy specific properties which ensure that the division algorithm can be applied efficiently. Specifically:

• Multiplicativity: \(d(ab) = d(a) + d(b)\) for all \(a, b \in R\).
• Triangle-like inequality: \(d(a+b) \leq \max \{d(a), d(b)\}\), with equality if \(d(a) eq d(b)\).
• Surjectivity: Every non-negative integer is achieved by \(d\), meaning any size within the range of the Euclidean function is possible to find within the domain.

A deep understanding of Euclidean functions is essential for comprehending how Euclidean domains facilitate similar operations as traditional numerical divisions.

Polynomial Representation

In a Euclidean domain \(R\), any nonzero element can uniquely be expressed in terms of other elements, much like how numbers are expressed in terms of powers of 10. Let's look at how this idea extends to polynomial representation.
If there's an element \(x\) in \(R\) such that \(d(x) = 1\), one can write any nonzero element \(a\) in \(R\) uniquely as:
\[ a = a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0 \]
where \(n = d(a)\), and each \(a_i\) belongs to a subfield \(F\) within \(R\). The coefficient \(a_n eq 0\) ensures the highest degree term does not cancel out.

• This framework is akin to expressing integers as sums of powers of 10, where polynomials are built up from powers of \(x\).
• Uniqueness holds because the Euclidean function helps maintain an order, preventing two different series from reducing to the same element.

Understanding this concept lets one simplify computations in the domain, performing operations like addition and multiplication with polynomial-like precision and structure.

Subfield

A subfield is a subset of a larger field that is itself a field under the same operations. The properties of a subfield are vital in the study of algebraic structures, offering a more granular view of how elements interact within these structures.
For a subset \(F = \{a \in R: d(a) \leq 0\}\) within a Euclidean domain \(R\), to be considered a subfield, it must be closed under three operations: addition, multiplication, and taking inverses excluding zero.

• Addition and Multiplication: If \(a, b \in F\), then \(d(a+b) \leq \max \{d(a), d(b)\} \leq 0\) and \(d(ab) = d(a) + d(b) \leq 0\). Thus, both operations produce results still inside \(F\).
• Inverses: Assuming \(a eq 0\), since \(d(a) \leq 0\) and given the Euclidean function's properties, there exists a \(b\) such that \(ab = 1\), ensuring \(b\) belongs to \(F\).

These conditions confirm \(F\) as a subfield, maintaining the familiar arithmetic properties but within the constraints of the Euclidean function's values.