Find the prime decomposition of $(2) $ in $\mathbb Z\left[ \frac{ 1 + \sqrt n }{2 } \right]$

Question

Let $n$ be an integer such that $ n \equiv 1 $ mod $4$.

Let $\mathbb Z \left[ \frac{ 1 + \sqrt n }{ 2} \right]$ be our ring. Let $(2)$ be the ideal generated by $2$.

What is the prime decomposition of $(2)$ in $\mathbb Z \left[ \frac{ 1 + \sqrt n }{ 2} \right]$?

Answer 1

Let ${\cal O}_K$ the ring of integers of $K=\Bbb Q(\sqrt{m})$ ($m$ any squarefree integer). Then the ideal $2{\cal O}_K$ factorizes as follows:

• $2{\cal O}_K=(2,\sqrt{m})^2$ if $2|m$
• $2{\cal O}_K=(2,1+\sqrt{m})^2$ if $m\equiv3\bmod4$,
• $2{\cal O}_K=(2,\frac{1+\sqrt{m}}2)(2,\frac{1-\sqrt{m}}2)$ if $m\equiv1\bmod8$,
• $2{\cal O}_K$ is prime if $m\equiv5\bmod8$.

Once these decompositions are given, proving them is a simple exercise.

Mind that sometimes the ideals in the decomposition may be actually principal (generated by one element). This obviously happen when ${\cal O}_K$ is a PID.