Problem 26 - Algebra: Pure and Applied

Question

Let \(d \in \mathbb{Z}\) be such that \(\sqrt{d} \notin \mathrm{Q} .\) In \(\mathbb{Z}[\sqrt{d}]\) let $$ v(z)=|z \bar{z}|=\left|a^{2}-d b^{2}\right|, \text { where } z=a+b \sqrt{d} $$ (a) (Pell's equation) Show that \(a+b \sqrt{d}\) is a unit in \(\mathbb{Z}[\sqrt{d}]\) if and only if \(a^{2}-d b^{2}=\pm 1\) (b) Assume that for any rational numbers \(x\) and \(y\) there exist integers \(n\) and \(m\) such that \(I(x-n)^{2}-d(y-m)^{2} \mid<1\). Show that in this case \(\mathbb{Z}[\sqrt{d}]\) is a Euclidean domain $$ \text { with } v(a+b \sqrt{d})=\left|a^{2}-d b^{2}\right| \text { . } $$

Step-by-Step Solution

Answer

(a) Units satisfy \(a^2 - db^2 = \pm 1\) (Pell's equation). (b) The given condition implies \(\mathbb{Z}[\sqrt{d}]\) is a Euclidean domain.

Step 1: Understanding the Problem Context

We are working with elements of the type \(a + b\sqrt{d}\) in the ring \(\mathbb{Z}[\sqrt{d}]\), where \(d\) is not a perfect square and \(a, b\) are integers. The norm of \(z\) is given by \(v(z) = |a^2 - db^2|\). The task involves characterizing units in this ring and understanding its Euclidean property.

Step 1: Establish Unit Criteria

Units in a ring \(R\) are elements \(z\) such that there exists a \(z'\) in \(R\) with \(zz' = 1\). For \(z = a + b\sqrt{d}\), consider its norm \(v(z) = |a^2 - db^2|\). For \(z\) to be a unit, \(v(z)\) must be 1, giving us the equation \(a^2 - db^2 = \pm 1\). This form is known as Pell's equation.

Step 2: Verify Conditions for a Euclidean Domain (Part b)

For \(\mathbb{Z}[\sqrt{d}]\) to be a Euclidean domain, a function \(v\) (as given) must satisfy the division algorithm: For all \(z, w eq 0\), there exist \(q, r\) such that \(z = qw + r\) with \(v(r) < v(w)\). The assumption provides that for any rational \(x, y\), there exist integers \(n, m\) such that \(v((x-n)^2 - d(y-m)^2) < 1\). This condition ensures reducing the norm while performing division steps on the elements.

Step 3: Apply Assumptions to Prove Euclidean Property

Given that we can approximate \((x - n)^2 - d(y - m)^2\) arbitrarily close to 0 (i.e., the absolute norm less than 1), the remainder \(r\) can be made smaller than \(v(w)\), the norm of any non-zero divisor \(w\). This demonstrates \(\mathbb{Z}[\sqrt{d}]\) admits the division properties characteristic of a Euclidean domain.

Key Concepts

Euclidean Domain

A Euclidean domain is a special kind of ring where you can perform a division algorithm similar to what you do with integers. In simpler terms, within a Euclidean domain, for any two elements, say \(z\) and \(w\) (where \(w eq 0\)), you can find two other elements \(q\) (quotient) and \(r\) (remainder) such that \(z = qw + r\) and the `norm` of \(r\) is less than the `norm` of \(w\). Why is this important? It allows mathematicians to perform unique factorization and division tasks that make the arithmetic particularly well-behaved. The norm acts like a measurement of size within the ring, aiding in these calculations. In the context of \(\mathbb{Z}[\sqrt{d}]\), having this property means it's possible to approximate solutions closely using rational numbers, thereby confirming \(\mathbb{Z}[\sqrt{d}]\) is a Euclidean domain.

Algebraic Numbers

Algebraic numbers are a broader category of numbers that include roots of polynomial equations with rational coefficients. This means they can be expressed as solutions to equations like \(x^n + a_{n-1}x^{n-1} + \ldots + a_0 = 0\), where all coefficients \(a_i\) are rational numbers. This concept is central in number theory as algebraic numbers extend our familiar realm of integers and rational numbers, enabling the construction of richer mathematical structures, such as fields and rings. For example, a number like \(\sqrt{d}\), where \(d\) is not a perfect square, is algebraic because it is a solution to the equation \(x^2 - d = 0\). Understanding algebraic numbers allows us to explore number systems where certain irrational numbers behave predictably, aiding in solving equations like Pell's equation seen in the problem context.

Norm in Algebraic Number Rings

The concept of a norm in algebraic number rings helps us understand how 'big' or 'small' an element is, establishing a method of comparing different elements' sizes. Specifically, in the ring \(\mathbb{Z}[\sqrt{d}]\), the norm of an element \(z = a + b\sqrt{d}\) is defined as \(v(z) = |a^2 - db^2|\). This norm has significant implications:

• It determines whether or not an element is a unit. If \(v(z) = 1\), \(z\) is a unit, satisfying the equation \(a^2 - db^2 = \pm 1\), a form of Pell's equation.
• It plays a vital role in establishing whether the ring is a Euclidean domain. As it defines division processes, it also helps evaluate how close an element is to being a multiple of another.

A firm grasp of how norms function in the algebraic number context broadens a student's capacity to work with more diverse and complex number systems, particularly those stemming from irrational solutions.