First of all, it is clear that $\Z[\sqrt{2}]$ is an integral domain since it is contained in $\R$.
We use the norm given by the absolute value of field norm.
Namely, for each element $a+\sqrt{2}b\in \Z[\sqrt{2}]$, define
\[N(a+\sqrt{2}b)=|a^2-2b^2|.\]
Then the map $N:\Z[\sqrt{2}] \to \Z_{\geq 0}$ is a norm on $\Z[\sqrt{2}]$.
Also, it is multiplicative:
\[N(xy)=N(x)N(y).\]
Remark that since this norm comes from the field norm of $\Q(\sqrt{2})$, the multiplicativity of $N$ holds for $x, y \in \Q(\sqrt{2})$ as well.
We show the existence of a Division Algorithm as follows.
Let
\[x=a+b\sqrt{2} \text{ and } y=c+d\sqrt{2}\]
be arbitrary elements in $\Z[\sqrt{2}]$, where $a,b,c,d\in \Z$.
We have
\begin{align*}
\frac{x}{y}=\frac{a+b\sqrt{2}}{c+d\sqrt{2}}=\frac{(ac-2bd)+(bc-ad)\sqrt{2}}{c^2-2d^2}=r+s\sqrt{2},
\end{align*}
where we put
\[r=\frac{ac-2bd}{c^2-2d^2} \text{ and } s=\frac{bc-ad}{c^2-2d^2}.\]
Let $n$ be an integer closest to the rational number $r$ and let $m$ be an integer closest to the rational number $s$, so that
\[|r-n| \leq \frac{1}{2} \text{ and } |s-m| \leq \frac{1}{2}.\]
Let
\[t:=r-n+(s-m)\sqrt{2}.\]
Then we have
\begin{align*}
t&=r+s\sqrt{2}-(n+m\sqrt{2})\\
&=\frac{x}{y}-(n+m\sqrt{2}).
\end{align*}
It follows that
\begin{align*}
yt=x-(n+m\sqrt{2})y \in \Z[\sqrt{2}].
\end{align*}
Thus we have
\begin{align*}
x=(n+m\sqrt{2})y+yt \tag{*}
\end{align*}
with $n+m\sqrt{2}, yt\in \Z[\sqrt{2}]$.
We have
\begin{align*}
N(t)&= |(r-n)^2-2(s-m)^2|\\
&\leq |r-n|^2+2|s-m|^2\\
& \leq \frac{1}{4}+2\cdot\frac{1}{4}=\frac{3}{4}.
\end{align*}
It follows from the multiplicativity of the norm $N$ that
\begin{align*}
N(yt)=N(y)N(t)\leq \frac{3}{4}N(y)< N(y).
\end{align*}
Thus the expression (*) gives a Division Algorithm with quotient $n+m\sqrt{2}$ and remainder $yt$.
Related Question.
Problem. In the ring $\Z[\sqrt{2}]$, prove that $5$ is a prime element but $7$ is not a prime element.
The Quotient Ring $\Z[i]/I$ is Finite for a Nonzero Ideal of the Ring of Gaussian Integers
Let $I$ be a nonzero ideal of the ring of Gaussian integers $\Z[i]$.
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Proof.
Recall that the ring of Gaussian integers is a Euclidean Domain with respect to the norm
\[N(a+bi)=a^2+b^2\]
for $a+bi\in \Z[i]$.
In particular, […]
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\[N(\alpha)=\alpha \bar{\alpha}=a^2+b^2.\]
Here $\bar{\alpha}$ is the complex conjugate of […]
5 is Prime But 7 is Not Prime in the Ring $\Z[\sqrt{2}]$
In the ring
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show that $5$ is a prime element but $7$ is not a prime element.
Hint.
An element $p$ in a ring $R$ is prime if $p$ is non zero, non unit element and whenever $p$ divide $ab$ for $a, b \in R$, then $p$ […]
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Determine all the ring homomorphisms from $\Z$ to $R$.
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For each positive integer $n$, prove that the polynomial
\[(x-1)(x-2)\cdots (x-n)-1\]
is irreducible over the ring of integers $\Z$.
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Prove that the quadratic integer ring $\Z[\sqrt{5}]$ is not a Unique Factorization Domain (UFD).
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Let $R$ be a commutative ring with $1$. Show that if $R$ is an integral domain, then the characteristic of $R$ is either $0$ or a prime number $p$.
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You write: “It follows that $yt=x-(n+m\sqrt{2})\in \Z[\sqrt{2}]$.” I think you dropped a factor of $y$. Shouldn’t it be $yt=x-(n+m\sqrt{2})y^2 \in \Z[\sqrt{2}]? I think this might affect the rest of your solution.
Thank you for catching a typo. I modified the proof.
(Previously, there was an extra “$y$” in the expression for $t$ and one $y$ is missing in the expression for $yt$.)
You write: “It follows that $yt=x-(n+m\sqrt{2})\in \Z[\sqrt{2}]$.” I think you dropped a factor of $y$. Shouldn’t it be $yt=x-(n+m\sqrt{2})y^2 \in \Z[\sqrt{2}]? I think this might affect the rest of your solution.
Dear Eli,
Thank you for catching a typo. I modified the proof.
(Previously, there was an extra “$y$” in the expression for $t$ and one $y$ is missing in the expression for $yt$.)