5 is Prime But 7 is Not Prime in the Ring $\Z[\sqrt{2}]$
Problem 224
In the ring
\[\Z[\sqrt{2}]=\{a+\sqrt{2}b \mid a, b \in \Z\},\]
show that $5$ is a prime element but $7$ is not a prime element.
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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 $p$ divides $a$ or $b$.
Equivalently, an element $p$ in the ring $R$ is prime if the principal ideal $(p)$ generated by $p$ is a nonzero prime ideal of $R$.
Proof.
5 is a prime element in the ring $\Z[\sqrt{2}]$.
We first show that $5$ is prime in the ring $\Z[\sqrt{2}]$.
Suppose that
\[5|(a+\sqrt{2}b)(c+\sqrt{2}d)\]
for $a+\sqrt{2}b, c+\sqrt{2}d \in \Z[\sqrt{2}]$.
By taking the norm, we obtain
\[25| (a^2-2b^2)(c^2-2d^2)\]
in $\Z$.
From this, we may assume that $5|a^2-2b^2$.
Now look at the following table.
\begin{array}{ |c|c|c|c| }
\hline
a, b & a^2, b^2 \pmod{5} & 2b^2 \pmod{5} \\
\hline
0 & 0 & 0 \\
1& 1 & 2 \\
2& 4 & 3 \\
3 & 4 & 3\\
4 & 1 & 2\\
\hline
\end{array}
From this table, we see that $a^2-2b^2=0 \pmod{5}$ if and only if $a, b$ are both divisible by $5$.
Therefore $5|a+\sqrt{2}b$, and $5$ is a prime element in $\Z[\sqrt{2}]$.
7 is not a prime element in the ring $\Z[\sqrt{2}]$.
Next, we show that $7$ is not a prime element in $\Z[\sqrt{2}]$.
To see this, note that we have
\[7=(3+\sqrt{2})(3-\sqrt{2})\]
and $7$ does not divide $3+\sqrt{2}$ and $3-\sqrt{2}$.
Hence $7$ is not a prime element in the ring $\Z[\sqrt{2}]$.
Related Question.
For a proof of this fact, see that post “The Ring $\Z[\sqrt{2}]$ is a Euclidean Domain“.
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