Proving $\mathbb{Z}$, $\mathbb{Z}$, $\mathbb{Z}$, and $\mathbb{Z}$ are euclidean.

I have this short class note from my graduate number theory:


THEOREM: Assume that $\vert N(x + y \sqrt d)\vert < 1$ for any two rational numbers $x$ and $y$ with $\vert x \vert \leq 1/2$ and $\vert y \vert \leq 1/2$. Define $\delta : \mathbb Z[\sqrt d] \setminus \{0\} \to \mathbb N$ by $z \mapsto \vert N(z)\vert$, then $\mathbb Z[\sqrt d]$ is euclidean with regards to $\delta$.

PROOF: Omitted.

COROLLARY: The integral domains $\mathbb{Z}\left[\sqrt{-2}\right]$, $\mathbb{Z}\left[\sqrt{-1}\right]$, $\mathbb{Z}\left[\sqrt{2}\right]$, and $\mathbb{Z}\left[\sqrt{3}\right]$ are euclidean.

PROOF: Let x and y be rational numbers with $|x| \leq 1/2$ and $|y| \leq 1/2$. Then

$$|N\left(x + y\sqrt{-2}\right)| = |x^2 + 2y^2| \leq 3/4 < 1,$$

$$|N\left(x + y\sqrt{-1}\right)| = |x^2 + y^2| \leq 1/2 < 1,$$

$$|N\left(x + y\sqrt{2}\right)| = |x^2 – 2y^2| \leq 1/2 < 1,$$

$$|N\left(x + y\sqrt{3}\right)| = |x^2 – 3y^2| \leq 3/4 < 1.$$

This proves the corollary.


My questions are about the Corollary:

  1. Why did my professor make the assumption that $|x| \leq 1/2$ and $|y| \leq 1/2$, when it is about $\mathbb{Z}$ the integer? (Post Script: Never mind about this question, I got it now from the Theorem before the Corollary.)
  2. There should be missing explanations before he suddenly jumped to “This proves the corollary.” What are they?

Any help would be very much appreciated. Thank you for your time.

Solutions Collecting From Web of "Proving $\mathbb{Z}$, $\mathbb{Z}$, $\mathbb{Z}$, and $\mathbb{Z}$ are euclidean."

For (1), this is a question about the theorem you mentioned, it has to do with choosing the integer closest to the rational number you chose. For any rational number you can choose the closest integer to it, and it will be less than or equal to $\displaystyle \frac12$ away.

You end up choosing $\alpha, \beta \in \mathbb Z[\sqrt d]$ where $\beta \neq 0$ and then consider $\displaystyle \frac\alpha\beta$ which can be written as $r + s \sqrt d$ where $r, s \in \mathbb Q$, then choose $x, y$ integers such that $x$ is closest integer to $r$ and $y$ is closest integer to $s$.

For (2), you have the theorem at your disposal that says if (some hypothesis are satisfied on $\mathbb Z[\sqrt d]$) then $\mathbb Z[\sqrt d]$ is a Euclidean domain (The proof of that theorem should somehow show it is Euclidean, along the lines of $a = bq + r$ with $r = 0$ or $N(r) < N(b)$.)

In your corollary, you show that each of $\mathbb Z[\sqrt -2], \mathbb Z[\sqrt -1], \mathbb Z[\sqrt 2], \mathbb Z[\sqrt 3]$ satisfy the given hypothesis, hence you can conclude that each of those are Euclidean domains.