Question

Consider the ring \(R=Z[1 / 2\\}=\left\\{a / 2^{n}: a \in Z_{n} n \in \mathbb{N}\right\\}\) of binary rationals. (i) Prove that \(R\) is the smallest subring of \(Q\) containing \(\mathrm{Z}\) and 1/2. (ii) What are the units of \(R\) ? (iii) You may use the fact that \(R\) is a UFD and that any two elements of \(R\) have a ged which is unique up to associates. Find a normal form on \(R\) and use this to define a ged function on \(R\). (iv) Determine the content and primitive part of the polynomial \(f=2 x^{2}+6 x-4\) with respect to the three rings \(Z, R\), and \(Q\). Is \(f\) primitive with respect to \(R\) ?

Short answer

(i) Yes, (ii) Units are \(1\) and \(-1\), (iii) Normal form is \(\frac{b}{2^m}\), (iv) No, \(f\) is not primitive in \(R\).

Step-by-step solution

Understand the Definition of R

The ring \( R = \mathbb{Z}[\frac{1}{2}] = \{ \frac{a}{2^n} : a \in \mathbb{Z}, n \in \mathbb{N} \} \) consists of all rational numbers that can be expressed as integers divided by a power of 2. Thus, every element in \( R \) can be written in that form.

Prove R is the Smallest Subring

To prove that \( R \) is the smallest subring of \( \mathbb{Q} \) containing \( \mathbb{Z} \) and \( \frac{1}{2} \), notice that any subring of \( \mathbb{Q} \) containing \( \mathbb{Z} \) and \( \frac{1}{2} \) must contain all elements of the form \( \frac{1}{2}^n \) since subrings are closed under multiplication and addition. Therefore, \( R \) consisting of these elements is indeed the smallest.

Identify Units of R

The units of a ring are elements that have multiplicative inverses. In \( R \), an element \( \frac{a}{2^n} \) has an inverse if and only if \( a \) is \( \pm 1 \). Thus, the units in \( R \) are \( 1 \) and \( -1 \), as their inverses are also contained within \( R \).

Normal Form in a UFD

A unique factorization domain (UFD) ensures each element can be factored into irreducibles. Here, each \( x = \frac{a}{2^n} \in R \) can be written as \( \pm \frac{b}{2^m} \) where \( b \) has no common factors with 2. This form ensures uniqueness; hence it is a normal form.

Define GED Function on R

Given two elements \( \frac{a}{2^n}, \frac{b}{2^m} \) in \( R \), their greatest element divisor (GED) is determined by the gcd of their numerators and the least common multiple of their powers of 2. Therefore, \( \text{GED}(\frac{a}{2^n}, \frac{b}{2^m}) = \frac{\gcd(a, b)}{2^{\min(n, m)}} \).

Calculate Content and Primitive Part for Z, R, Q

The content of a polynomial is the gcd of its coefficients. For \( f = 2x^2 + 6x - 4 \), in \( \mathbb{Z} \), the content is 2, as the gcd of \( 2, 6, -4 \) is 2. In \( R \), the content is \( \frac{1}{2} \), as \( R \) allows division by powers of 2. In \( \mathbb{Q} \), the content is 1, as \( \mathbb{Q} \) factors out all common rational multiples. The primitive part, after dividing the polynomial by its content, remains the same for each ring when considering their respective contents: \( x^2 + 3x - 2 \).

Assess if f is Primitive with Respect to R

For a polynomial to be primitive over a ring, its content with respect to that ring must be 1. Since the content of \( f \) in \( R \) is \( \frac{1}{2} \), not 1, \( f \) is not primitive in \( R \).

Key concepts

Unique Factorization Domain

A Unique Factorization Domain (UFD) is a special type of integral domain where every non-zero, non-unit element can be factored uniquely into irreducible elements, much like prime numbers in the integers. This property makes UFDs particularly useful in algebraic number theory and polynomial factorization.

In a UFD, if you have an element, it can be expressed in only one way (up to the order of the factors and units) as a product of irreducible elements. This attribute ensures that the factorization of elements is predictable and uniform, aiding in clear mathematical computation and analysis.

For example, in the ring \( R = \mathbb{Z}[\frac{1}{2}] \), the element \( \frac{a}{2^n} \) can be presented in a specific form consistently. Being able to reduce elements to a normal form through factorization is crucial for defining concepts such as a normal form or greatest element divisor (GED) which are based on this principle.

Subring Properties

A subring, as the name suggests, is a subset of a larger ring that is itself a ring. To be considered a subring, a subset must satisfy certain properties: it must be closed under addition and multiplication, and it also must contain the multiplicative identity of the larger ring (if there is one).

• **Containment**: A subring must contain enough elements to make it stable under the ring operations.
• **Non-empty Set**: It must contain at least one element (often the zero of the ring).

In the example of the ring \( R = \mathbb{Z}[\frac{1}{2}] \), to prove that it's the smallest subring of \( \mathbb{Q} \) containing \( \mathbb{Z} \) and \( \frac{1}{2} \), you observe the closure properties. Any subring containing \( \mathbb{Z} \) and \( \frac{1}{2} \) must include all elements that can be formed by multiplying and adding these elements, essentially forming \( R \). This demonstrates its minimal structure, capturing precisely the necessary elements.

Polynomial Content

Polynomial content is essentially the greatest common divisor (GCD) of the coefficients of a polynomial. The concept is crucial in understanding the simplification and analysis of polynomials within different rings.

For a polynomial like \( f = 2x^2 + 6x - 4 \), the content depends on the coefficients' GCD within a particular ring. In \( \mathbb{Z} \), the content is 2 since the coefficients are integers. In \( R \), where denominators can be powers of 2, the content is \( \frac{1}{2} \). In \( \mathbb{Q} \), the content is 1, representing the ring's allowance for factoring out any rational number.

The process of determining if a polynomial is primitive (i.e., its content is 1 within its specific ring domain) involves looking at how these GCDs interact with the ring's structure and its elements.

Units in a Ring

Units in a ring are elements that have a multiplicative inverse within the same ring. Understanding units is crucial as it defines identities and inversibility within the algebraic structure.

In the ring \( R = \mathbb{Z}[\frac{1}{2}] \), a rational of the form \( \frac{a}{2^n} \) is a unit if there exists another element within \( R \) such that their product is 1. For \( \frac{a}{2^n} \) to have an inverse, \( a \) must be \( \pm 1 \), making these two numbers, \( 1 \) and \( -1 \), the only units in \( R \).

The presence of units affects how elements in a ring interact in multifaceted algebraic operations, influencing both the structure of algebraic expressions and their solutions.