Quadratic extensions rings

note

This lab was written by Elad Zelingher, all rights reserved to Elad Zelingher.

Let $\alpha$ be an integer that is not a square. We define $\mathbb{Z}[\sqrt{\alpha}]$ to be the following subset of the complex numbers

$$\begin{equation*} \mathbb{Z}[\sqrt{\alpha}] = \left\{ x + y \sqrt{\alpha} \mid x,y \in \mathbb{Z} \right\}. \end{equation*}$$

The set $\mathbb{Z}[\sqrt{\alpha}]$ is a ring: by this we mean that:

• $0,1 \in \mathbb{Z}[\sqrt{\alpha}]$.
• If two elements belong to $\mathbb{Z}[\sqrt{\alpha}]$ then their sum and difference also belong to $\mathbb{Z}[\sqrt{\alpha}]$.
• If two elements belong to $\mathbb{Z}[\sqrt{\alpha}]$ then their product also belongs to $\mathbb{Z}[\sqrt{\alpha}]$.

An element $u \in \mathbb{Z}[\sqrt{\alpha}]$ is called a unit if there exists $u' \in \mathbb{Z}[\sqrt{\alpha}]$ such that $u u' = 1$. In this case we call $u'$ the inverse of $u$.

An element $a \in \mathbb{Z}[\sqrt{\alpha}]$ is called irreducible if $a$ is not a unit and for every $b,c \in \mathbb{Z}[\sqrt{\alpha}]$ such that $a = bc$ we have that either $b$ or $c$ is a unit.

We will prove in class that every element $0 \ne a \in \mathbb{Z}[\sqrt{\alpha}]$ can be written in the form

$$\begin{equation*} a = \epsilon p_1 p_2 \dots p_{\ell}. \end{equation*}$$

where:

• $\epsilon$ is a unit.
• $p_i$ is a an irreducible element for every $1 \le i \le \ell$.

We will refer to the equation above as a factorization of $a$.

Problem 1

The app QuadraticRingFactorization allows you to see the different factorizations of elements in $\mathbb{Z}[\sqrt{\alpha}]$. By selecting the checkbox, you can restrict the list to display only prime numbers.

Use the app with $\alpha = -1$ and $n = 100$ and select "Show only prime numbers". What prime numbers remain irreducible in $\mathbb{Z}\left[i\right]?$ Can you identify a pattern?

Problem 2

Repeat the process above to identify a pattern for which prime numbers remain irreducible in $\mathbb{Z}\left[\sqrt{-3}\right]$.

Problem 3

Repeat the process above to identify a pattern for which prime numbers remain irreducible in $\mathbb{Z}\left[\sqrt{-2}\right]$.

If you are stuck, use The On-Line Encyclopedia of Integer Sequences (OEIS): make a list of the prime numbers that remain irreducible in $\mathbb{Z}\left[\sqrt{-2}\right]$ and enter that sequence to OEIS to see if a pattern is already known.

Problem 4

We say that $\mathbb{Z}\left[\sqrt{\alpha}\right]$ is a unique factorization domain if for every $a \ne 0$ the decomposition

$$\begin{equation*} a = \epsilon p_1 p_2 \dots p_{\ell}. \end{equation*}$$

is unique, up to possibly:

• Replacing $\epsilon$ with a different unit $\epsilon'$.
• Replacing $p_i$ with $p'_i = u_i p_i$ where $u_i$ is a unit.
• Reordering the $p_i$s.

Use QuadraticRingFactorization to check if $\mathbb{Z}[i]$ is a unique factorization domain? Is $\mathbb{Z}[\sqrt{-2}]$? What about $\mathbb{Z}[\sqrt{-3}]$, $\mathbb{Z}[\sqrt{2}]$ and $\mathbb{Z}[\sqrt{3}]$?

Problem 5

Is $\mathbb{Z}[\sqrt{5}]$ a unique factorization domain? Why?

Problem 6

Make a list of $\alpha$ for which $\mathbb{Z}[\sqrt{\alpha}]$ is a unique factorization domain and look it up on OEIS.