Prime elements in $\mathbb{Z}[\sqrt{2}]$

Question

What are the prime elements in the ring $\mathbb{Z}[\sqrt{2}]$?

Note that since the ring is a PID (and thus a UFD) then prime = irreducible. Even more, it is Euclidean with respect to the absolute value of the norm: $N(a+b\sqrt{2})=a^2-2b^2$ so it has a nice structure.

I know that if $N(\alpha)=p$ with $p$ prime integer then $\alpha$ is a prime element. Not a very explicit description but it would do.

Nevertheless, I think there are other primes not covered by the above case. What about prime integers, are they still prime in $\mathbb{Z}[\sqrt{2}]$? I am hoping a classification can be found similar to the primes in $\mathbb{Z}[i]$...

Edit: As Alex Youcis points out, it is probably useful to keep in mind that all the units in this ring are characterized by $\pm(1-\sqrt{2})^n$, so the search for prime elements should be up to these.

Answer 1

This is a partial (unsatisfyingly partial) answer:

There is a nice theorem in algebraic number theory:

Theorem(Dedekind): Let $\mathcal{O}_K$ be a number ring such that there exists $\alpha\in\mathcal{O}_K$ with $\mathcal{O}_K=\mathbb{Z}[\alpha]$. Let $f(T)$ be the minimal polynomial of $\alpha$ over $\mathbb{Q}$. Then, for every prime $p\in\mathbb{Z}$ let $\overline{g_1(T)}^{e_1}\cdots \overline{g_m(T)}^{e_m}=\overline{f(T)}$ be the prime factorization of $\overline{f(T)}\in\mathbb{F}_p[T]$ (where $g_i(T)\in\mathbb{Z}[T]$). Then, the prime ideals of $\mathcal{O}_K$ lying over $p$ are those of the form $(p,g_i(\alpha))$ for $i=1,\ldots,m$.

So, now we have the case when $K=\mathbb{Q}(\sqrt{2})$, $\alpha=\sqrt{2}$, and $f(T)=T^2-2$. Thus, the find the prime ideal of $\mathbb{Z}[\sqrt{2}]$ lying above $p$ we need only factor $T^2-2$ in $\mathbb{F}_p[T]$. Now, by quadratic reciprocity we know that $2$ has a square root in $\mathbb{F}_p$ if and only if $p\equiv 1,7\mod 8$.

If $p\equiv 3,5\mod 8$ then $T^2-2$ is irreducible in $\mathbb{F}_p[T]$ and thus the prime ideal lying above $p$ is $(p,(\sqrt{2})^2-2)=(p)$.

If $p\equiv 1,7\mod 8$ then we know that $T^2-2$ factors in $\mathbb{F}_p[T]$ as $(T-\beta)(T+\overline{\beta})$ where $\beta$ is some (lift to $\mathbb{Z}$ of a) square root of $2$ in $\mathbb{F}_p$. Thus, Dedekind's theorem tells us that the prime ideals of $\mathbb{Z}[\sqrt{2}]$ lying above $p$ are $(p,\sqrt{2}\pm\beta)$. Now, abstractly we know then that $(p,\sqrt{2}\pm\beta)=\text{gcd}(p,\sqrt{2}\pm\beta)$, and that this gcd can be computed (in theory) using the Euclidean algorithm.

Thus, up to units of $\mathbb{Z}[\sqrt{2}]$ (which are all of the form $\pm(1-\sqrt{2})^n$, $n\in\mathbb{Z}$) the prime elements of $\mathbb{Z}[\sqrt{2}]$ are $p$ for $p\equiv 3,5\mod 8$, and $\text{gcd}(p,\sqrt{2}\pm\beta)$ where $\beta$ is a lift of a square root of $2$ in $\mathbb{F}_p$.