Primality Testing in $\mathcal{O}_{\mathbb{Q}(\sqrt{d})}$?

Question

What is known about the computational complexity of primality testing in $\mathcal{O}_{\mathbb{Q}(\sqrt{d})}$ where $d$ is a square-free number? For what values of $d$ is primality testing easy (i.e., can be determined in polynomial time)?

Answer 1

For $K = \mathbb{Q}(\sqrt{d})$, primality testing in $\mathcal{O}_K$ reduces to primality testing of integers. It's known that the prime elements $\alpha \in \mathcal{O}_K$ are either

• elements whose norms $N(\alpha)$ are prime, or
• elements whose norms $N(\alpha)$ are squares of a prime $p$ such that $\left( \frac{d}{p} \right) = -1$.

(A slight modification is necessary in the case that $d \equiv 1 \bmod 4$ and $N(\alpha) = 4$. In that case $\alpha$ is prime if and only if $\frac{d-1}{4}$ is odd.)

The first condition can be tested for in polynomial time by the AKS primality test. The second condition can be tested for in polynomial time by AKS and quadratic reciprocity.