Problem 68 - A Computational Introduction to Number Theory and Algebra

Question

Consider the subring \(\mathbb{Z}[1 / 2]\) of \(\mathbb{Q} .\) Show that \(\mathbb{Z}[1 / 2]=\left\\{a / 2^{i}:\right.\) \(a, i \in \mathbb{Z}, i \geq 0\\},\) that \((\mathbb{Z}[1 / 2])^{*}=\left\\{2^{i}: i \in \mathbb{Z}\right\\},\) and that every non-zero ideal of \(\mathbb{Z}[1 / 2]\) is of the form \((m),\) for some uniquely determined, odd integer \(m .\)

Step-by-Step Solution

Answer

The set of units in the subring \(\mathbb{Z}[1/2]\) is \(\{2^i: i \in \mathbb{Z}\}\).

Step 1: Show that \(\mathbb{Z}[1/2]=\{a/2^i: a,i \in \mathbb{Z}, i \geq 0\}\)

Recall that \(\mathbb{Z}[1/2]\) is the subring formed by taking \(\mathbb{Z}\) and adjoining the element \(1/2\). Thus, all elements of this subring are of the form \(a + b/2\), where \(a\) and \(b\) are integers. By multiplying the fraction with the necessary power of two, we can rewrite such elements as fractions with a denominator which is a power of two: \(a/2^i\), where \(a \in \mathbb{Z}\) and \(i \in \mathbb{Z}\), \(i\geq 0\). Therefore, \(\mathbb{Z}[1/2]=\{a/2^i: a,i \in \mathbb{Z}, i \geq 0\}\).

Step 2: Show that \((\mathbb{Z}[1/2])^*=\{2^i: i \in \mathbb{Z}\}\)

We need to find the set of units in the subring \(\mathbb{Z}[1/2]\). A unit is an element that has a multiplicative inverse. Let the element \(a/2^i \in \mathbb{Z}[1/2]\). We need to find \(b/2^j\) such that their product is \(1\): $$\frac{a}{2^i}\cdot\frac{b}{2^j} = 1 \iff ab = 2^{i+j}$$ Here, \(a\), \(b\) and \(i\) are integers, and \(i \geq 0\). The units are precisely those elements \(a/2^i\) for which there exists such a \(b/2^j\). Only powers of two can be multiplied by other powers of two to give a power of two. Thus, the set of units must be the powers of two: \(\{2^i: i \in \mathbb{Z}\}\).

Step 3: Show that every non-zero ideal of \(\mathbb{Z}[1/2]\) is of the form \((m)\), with \(m\) a uniquely determined odd integer

Let \(\mathcal{I}\) be a nonzero ideal of \(\mathbb{Z}[1/2]\). We want to show that \(\mathcal{I}=(m)\) for some \(m\) odd. First, let \(m\) be the smallest positive odd integer in the ideal \(\mathcal{I}\). Since \(\mathcal{I}\) is an ideal, it must be closed under multiplication, and we know that there is an \(x \in \mathbb{Z}[ 1 / 2 ]\) with \(m x \in \mathcal{I}\). Then, for any element \(n \in \mathcal{I}\), we can write \(n = m x + r\), where \(0 \leq r < m\). If \(r\) is odd, we get a contradiction, as we assumed that \(m\) is the smallest odd integer in the ideal. Therefore, \(r\) must be even. Dividing \(r\) by the highest power of two, we get an element in the form \(b/2^j\) with \(b\) odd. Since the ideal is closed under addition, \(b/2^j\) must be in the ideal. By the previous step, we know that \(2^j \in (\mathbb{Z}[1/2])^*\). So, multiplying \(b/2^j\) by the inverse of \(2^j\), we get that \(b \in \mathcal{I}\). But \(b\) is less than \(m\), which is a contradiction. Thus, \(r\) must be zero, and we have \(n = m x\). In this case, we showed that \(\mathcal{I}=(m)\) for a uniquely determined odd integer \(m\).

Key Concepts

Subrings

In algebraic number theory, a subring is a subset of a ring that is itself a ring with the same operations of addition and multiplication. Understanding subrings is crucial because it helps in the study of the structure and properties of numbers. In this context, we look at the subring \( \mathbb{Z}[1/2] \), which is the subring of the rational numbers \( \mathbb{Q} \) formed by adjoining \( 1/2 \) to \( \mathbb{Z} \). Every element in this subring has the form \( a/2^i \), where \( a \) and \( i \) are integers and \( i \geq 0 \). This form illustrates how elements can be expressed as fractions with powers of two in the denominator.

• Characteristics: Subrings must contain the multiplicative identity (which is 1 in this case), and they must be closed under subtraction and multiplication.
• Purpose: They play a role in constructing larger algebraic structures by building upon existing rings in a controlled manner.

Recognizing subrings involves ensuring these elements satisfy the rules of closure, identity, and inverse under ring operations. This understanding lays the foundation for exploring more complex structures in algebra.

Units in a Ring

Units in a ring refer to the elements that have a multiplicative inverse. These elements are essential because they maintain the structure of the ring when performing multiplication. In the case of the subring \( \mathbb{Z}[1/2] \), the units are found by identifying elements that can multiply to give the identity element, 1.
For example, if an element of the form \( a/2^i \) in \( \mathbb{Z}[1/2] \) has a multiplicative inverse, there exists another element \( b/2^j \) such that their product is 1:\[ \frac{a}{2^i} \cdot \frac{b}{2^j} = 1 \Rightarrow ab = 2^{i+j} \]

• Key Insight: Only powers of two can have inverses in this ring, meaning the units are \( 2^i \) where \( i \) varies over all integers. These powers form the set of units \( \{2^i: i \in \mathbb{Z}\} \).
• Practical Use: Understanding units allows one to grasp how division works in a ring and how cancellation properties are preserved.

By recognizing units, you can better understand how to manipulate and solve equations within the ring.

Ideals in Rings

An ideal in a ring is a subset in which the multiplication of its elements with any element from the ring results in an element also contained in the subset. Ideals help in understanding how rings can be divided or factored. With non-zero ideals in \( \mathbb{Z}[1/2] \), every ideal is of the form \((m)\), where \(m\) is a uniquely determined odd integer.
Here's how it works:

• Identify the smallest positive odd integer \(m\) within the ideal \(\mathcal{I}\).
• Elements of the ideal can be expressed as \(mx\), where \(x\) is in \( \mathbb{Z}[1/2] \).
• Proves that \( m \) fully generates the ideal because any other element can be reduced to a multiple of \( m \).

Uniqueness Insight: The uniqueness of \( m \) as an odd integer ensures that any element \( n \) in the ideal \( \mathcal{I} \) can be expressed without remainder.
Understanding ideals aids in the study of ring homomorphisms and quotient rings, providing a path to breaking down complex ring structures into manageable pieces.