Prove that $\mathcal{O}$ is a Euclidean Domain.

Question

Here is the question I am trying to solve (Dummit & Foote, 3rd edition, Chapter 8, section 1, #8(a)):

Let $F = \mathbb Q(\sqrt{D})$ be a quadratic field with associated quadratic integer ring $\mathcal{O}$ and field norm $N$ as in Section $7.1.$
$(a)$ Suppose $D$ is $-1,-2, -3, -7$ or $-11.$ Prove that $\mathcal{O}$ is a Euclidean domain with respect to $N.$

Here is the hint given to us in the book:

Modify the proof for $\mathbb Z[i]$ in the text. For $D = -3,-7,-11$ prove that every element of $F$ differs from an element in $\mathcal O$ by an element whose norm is at most $\frac{(1 + |D|)^2}{16 |D|},$ which is less than $1$ for these values of $D.$ Plotting the points of $\mathcal O$ in $\mathbb C$ maybe helpful.

And here is the proof that $\mathbb Z[i]$ is a Euclidean domain (from John Beachy, 4th edition):

Still exactly I do not know how to modify this proof to proof the required, could anyone show me the details of the proof of at least one case please?

Also, I found the question below here but I do not know if it is related or not

Show that the quadratic integer ring $\mathcal{O}=\{a+b\frac{1+\sqrt{-3}}{2}|a, b\in\mathbb{Z}\}$ is an Euclidean Domain.

Answer 1

I cannot give you the details in one of the cases without completely solving the exercise for you. Instead, allow me to split the proof up into parts, and you can see where you have difficulty adapting it.

1. Instead of $n+mi$, we have $a=n+m\sqrt{D}$, similarly $b=s+t\sqrt{D}$.
2. Argue that the quotient can still be written on the form $x+y\sqrt{D}$, but with $x,y\in\Bbb{R}$.
3. Find the best approximant $q$ of $x+y\sqrt{D}$ in $\mathcal{O}$. Note that this is the critical step, where you are using the concrete values of $D$ given. If $D=-2$, the completely same choice as in the proof for $\Bbb{Z}[i]$ works. If $D=-3,-7,-11$, you need to use that you know the exact form of a general element of $\mathcal{O}$, so your choice can be slightly better.
4. Compute the norm of $r=a-bq$ in the same fashion, and use the bounds you got in choosing the approximant $q$ to show that it is less than the norm of $b$.