Problem 35 - A Computational Introduction to Number Theory and Algebra

Question

Consider the polynomial $$ X^{3}-1=(X-1)\left(X^{2}+X+1\right) $$ Over \(\mathbb{C},\) the roots of \(X^{3}-1\) are \(1,(-1 \pm \sqrt{-3}) / 2 .\) Let \(\omega:=(-1+\sqrt{-3}) / 2,\) and note that \(\omega^{2}=-1-\omega=(-1-\sqrt{-3}) / 2,\) and \(\omega^{3}=1\). (a) Show that the ring \(\mathbb{Z}[\omega]\) consists of all elements of the form \(a+b \omega,\) where \(a, b \in \mathbb{Z},\) and is an integral domain. This ring is called the ring of Eisenstein integers. (b) Show that the only units in \(\mathbb{Z}[\omega]\) are \(\pm 1, \pm \omega,\) and \(\pm \omega^{2}\). (c) Show that \(\mathbb{Z}[\omega]\) is a Euclidean domain.

Step-by-Step Solution

Answer

To summarize the solution: a) The ring \(\mathbb{Z}[\omega]\) consists of all elements of the form \(a+b \omega,\) with \(a, b \in \mathbb{Z}\), and is shown to be an integral domain by proving that it is closed under addition and multiplication, and that there are no non-zero divisors. b) The only units in \(\mathbb{Z}[\omega]\) are \(\pm 1, \pm \omega,\) and \(\pm \omega^{2}\), found by examining cases that satisfy the condition of a norm equal to 1. c) \(\mathbb{Z}[\omega]\) is a Euclidean domain, as shown by proving the existence of a function (the norm) that satisfies the required properties for the division algorithm.

Step 1: (a) Formation of the Ring and Integral Domain

To show that \(\mathbb{Z}[\omega]\) consists of all elements of the form \(a+b \omega,\) where \(a, b \in \mathbb{Z},\) and is an integral domain, we start by showing that it is closed under addition and multiplication. Given two elements \(a+b\omega\) and \(c+d\omega\) in \(\mathbb{Z}[\omega]\) such that \(a, b, c, d \in \mathbb{Z}\), we can perform addition and multiplication as follows: Addition: \((a+b\omega) + (c+d\omega) = (a+c) + (b+d)\omega\) Multiplication: \((a+b\omega)(c+d\omega) = ac + ad\omega + bc\omega + bd\omega^2 = (ac - bd) + (ad + bc)\omega\) As the results of addition and multiplication belong to the form \(a+b\omega\), \(\mathbb{Z}[\omega]\) is closed under these operations. It also has a commutative, associative, and distributive nature with respect to addition and multiplication. In addition, it contains a multiplicative identity \(1\) and additive identity \(0\). So, \(\mathbb{Z}[\omega]\) forms a ring. In order to show that it is an integral domain, we need to show that there are no non-zero divisors. Let's assume \(p = a+b\omega\) and \(q = c+d\omega\) are two non-zero elements of \(\mathbb{Z}[\omega]\) such that \(pq = 0\). Now, let's use the norm \(N(p) = a^2 + ab + b^2\). Then \(N(pq) = N(p)N(q) = 0\), implying either one of \(N(p), N(q)=0\). This is a contradiction to the fact \(p, q \neq 0\), hence \(\mathbb{Z}[\omega]\) is an integral domain.

Step 2: (b) Finding Units of \(\mathbb{Z}[\omega]\)

To find the units of \(\mathbb{Z}[\omega]\), we need to use the fact that an element is a unit if its norm is \(1\). Let's assume \(p = a+b\omega\) is a unit of \(\mathbb{Z}[\omega]\). Then its norm \(N(p) = a^2 + ab + b^2 = 1\). Now, there are six possible cases for \(a\) and \(b\): 1. \(a=1\), \(b=0\): \(p=1\), 2. \(a=-1\), \(b=0\): \(p=-1\), 3. \(a=0\), \(b=1\): \(p=\omega\), 4. \(a=0\), \(b=-1\): \(p=-\omega\), 5. \(a=-1\), \(b=1\): \(p=-1+\omega = \omega^2\), 6. \(a=1\), \(b=-1\): \(p=1-\omega = -\omega^2\). Therefore, the units of \(\mathbb{Z}[\omega]\) are \(\pm 1, \pm \omega,\) and \(\pm \omega^{2}\).

Step 3: (c) Euclidean Domain

We need to show that \(\mathbb{Z}[\omega]\) is a Euclidean domain. To do this, we will show that a function exists that satisfies the required properties. Let's use the norm function \(N(a+b\omega) = a^2 + ab + b^2\) defined before. We will show that for any two non-zero elements \(a, b \in \mathbb{Z}[\omega]\), there exist \(q,r \in \mathbb{Z}[\omega]\) such that: \(a = bq + r\), and \(N(r) < N(b)\) or \(r=0\). Starting with \(a, b \in \mathbb{C}\) such that \(b \neq 0\), let's consider \(\alpha = a/b \in \mathbb{C}\). We can find two integers \(q_1, q_2 \in \mathbb{Z}\) such that \(|\alpha - q_1 - q_2\omega| \leq \sqrt{3}/2\). Thus, \(q=q_1+q_2\omega \in \mathbb{Z}[\omega]\) and we define \(r=a - bq\). Then the remainder \(r \in \mathbb{Z}[\omega]\) as well. Finally, as per the definition of the remainder and the fact that \(|\alpha - q_1 - q_2\omega| \leq \sqrt{3}/2\), we get that \(|r| = |a - bq| = |b(\alpha - q)| \leq \sqrt{3}/2 |b|\). Thus, \(N(r) < N(b)\) or \(r=0\). Since the given conditions are satisfied, \(\mathbb{Z}[\omega]\) is a Euclidean domain.

Key Concepts

Integral Domain

An integral domain, in its essence, is a special type of ring which eliminates the possibility of zero divisors. This means that within an integral domain, the product of any two non-zero elements is always non-zero. This concept is crucial in the realm of abstract algebra as it provides a framework similar to that of ordinary arithmetic.

Let's reflect on the ring of Eisenstein integers, denoted as \(\mathbb{Z}[\omega]\). When we state that \(\mathbb{Z}[\omega]\) is an integral domain, we are essentially saying that if you take any two Eisenstein integers, let's name them \(p\) and \(q\), neither of which is zero, their product \(pq\) will also be distinctly different from zero. This characteristic ensures that the algebraic structure of Eisenstein integers is robust and eliminates certain complications that can arise in rings with zero divisors.

An additional aspect to note is that to demonstrate this for \(\mathbb{Z}[\omega]\), the solution used a norm function, \(N(p) = a^2 + ab + b^2\), which assigns a positive integer to each nonzero element, ensuring we can't have \(N(pq) = 0\) unless one of \(p\) or \(q\) is zero. In sum, the ring of Eisenstein integers isn't just a playground with regular numbers, but a sophisticated terrain that follows rules ensuring stable algebraic behavior akin to that of integers.

Euclidean Domain

Diving deeper into the structure of \(\mathbb{Z}[\omega]\), we come to understand that it's a Euclidean domain. So, what makes a Euclidean domain special? It is a type of integral domain that is equipped with an additional feature known as a Euclidean function. This function assigns to every non-zero element of the domain an integer in such a manner that, given any two elements \(a\) and \(b\), where \(b\) is non-zero, we can find two other elements, \(q\) and \(r\), so that \(a = bq + r\) with \(N(r) < N(b)\) or \(r=0\).

For Eisenstein integers, our Euclidean function is the norm \(N\) that we have used before. During your exploration of the problem solution, you may have noticed how this property helps in the division process, mirroring the way we perform division in our basic arithmetic with integers. The significance of being a Euclidean domain is that it makes many algebraic procedures more manageable, for example, finding the greatest common divisor of two elements, something which is integral in solving polynomial equations or factoring expressions within the domain.

The assurance that division will always work out nicely, without any messy leftovers, is what makes the ring of Eisenstein integers not just a beautifully structured set, but also a practically useful tool for mathematicians. It means algorithms that are applicable for integers and polynomials can also be applied to \(\mathbb{Z}[\omega]\), making the ring not only algebraically fascinating but also rich with applicability.

Polynomial Roots

When we talk about polynomial roots in the context of Eisenstein integers, we're peeking into the solutions of polynomial equations where the Eisenstein integers come into play. Specifically, the exercise you've worked on involves the roots of \(X^{3}-1\) in the complex plane, demonstrating that the roots include \(1\), and a pair of complex numbers often represented using \(\omega\) and its square.

In the context of Eisenstein integers, \(\omega\) is not just any number, but it's one of the complex roots of unity, besides 1, that satisfies \(\omega^{3} = 1\). This concept is closely linked with the geometric representation of numbers in the complex plane as \(\omega\) corresponds to a \(\frac{-1}{2}+ \frac{\sqrt{3}}{2}i\) which geometrically is a 120-degree rotation of the complex number 1. Further, it is worth noting that the exercise solution showed that the ring \(\mathbb{Z}[\omega]\) forms an integral domain and a Euclidean domain by showing that the elements, when expressed as polynomials in \(\omega\), have certain algebraic properties.

The roots of a polynomial also have deep implications in number theory and algebra, as they can shed light on the structure of numbers and offer solutions to various mathematical problems. In this case, the roots such as \(\omega\) offer a glimpse into the symmetrical and cyclical nature of certain algebraic systems.