Norms in 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. Recall that we defined $\mathbb{Z}[\sqrt{\alpha}]$ 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 defined the conjugate map $\sigma \colon \mathbb{Z}[\sqrt{\alpha}] \to \mathbb{Z}[\sqrt{\alpha}]$ by the rule $\sigma\left(x + y \sqrt{\alpha} \right) = x - y \sqrt{\alpha}$ for any $x,y \in \mathbb{Z}$. We also defined the norm map $\mathrm{N} \colon \mathbb{Z}[\sqrt{\alpha}] \to \mathbb{Z}$ by the rule $\mathrm{N}(z) = z \sigma(z)$. Explicitly, $\mathrm{N}\left(x + y \sqrt{\alpha}\right) = x^2 - \alpha y^2$. Recall that $\mathrm{N}(z_1 z_2) = \mathrm{N}(z_1) \mathrm{N}(z_2)$.

Problem 1

Recall that we showed that $\mathrm{N}\left(z\right) = \pm 1$ for $z \in \mathbb{Z}[\sqrt{\alpha}]$ if and only if $z$ is a unit. Try to prove this again.

Problem 2

The app QuadraticRingElements allows you to see the elements of $\mathbb{Z}[\sqrt{\alpha}]$ in a certain range among with their norm and whether they are irreducible or not. You may sort the elements by their norm or by the absolute value of their norm.

Use the app for different values of alpha such as $-1,-2,-3,-5,2,3,5$ and sort the elements by the absolute value of their norm. Notice that for any norm value there are usually four very similar elements with that norm. Can you explain why?

Problem 3

Use QuadraticRingElements with different values of $\alpha$ as above. For each of these values consider the elements with the smallest possible norm (in absolute value) that is not $\pm 1$ or $0$. What do these elements have in common? Can you prove this?

Problem 4

Again use QuadraticRingElements with different values of $\alpha$ as above. Now consider elements that their norm is a prime number (in absolute value). What do these elements have in common? Can you prove this property? Is the converse true?

Problem 5

We will prove soon in class that for every $a \ne 0$ not invertible there exists a decomposition

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

where $p_1$, $\dots$, $p_{\ell}$ are irreducible elements.

We say that $\mathbb{Z}\left[\sqrt{\alpha}\right]$ is a unique factorization domain if this decomposition is unique, up to possibly:

• Replacing $p_i$ with an associate, that is, with $p'_i = u_i p_i$ where $u_i$ is a unit.
• Reordering the $p_i$s.

Use QuadraticRingFactorization to check if $\mathbb{Z}[\sqrt{\alpha}]$ is a unique factorization domain for each $\alpha = -5,-3,-2,-1,2,3,5$.

Problem 6

For each $\alpha$ from the previous problem, use QuadraticRingFactorization and select the checkbox to display only prime numbers. What are the possible lengths for a decomposition of a prime number $p \in \mathbb{Z}$? Try to prove this.

Problem 7

Use QuadraticRingElements again for the values above, and sort by the absolute value of the norm. Check for small prime numbers $p = 2,3,5,7,\dots$ whether there exist an element such that $\mathrm{N}(z) = \pm p$. Make two lists, one for the prime numbers $p$ that are norms (in absolute value), and one for those that are not. Now use QuadraticRingFactorization and look at the decomposition of each $p$ you tested. Can you find a relation between the decomposition of $p$ and whether it is a norm (in absolute value) or not? Can you prove this?