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)?
Primality Testing in $\mathcal{O}_{\mathbb{Q}(\sqrt{d})}$?
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number-theory
computational-complexity
primality-test
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1$\mathbb{Z}[\sqrt{d}]$ isn't the full ring of integers if $d \equiv 1 \bmod 4$. – 2011-12-19
1 Answers
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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.
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1@Siddharth: probably this result can be found in most textbooks on algebraic number theory, possibly as an exercise. – 2014-01-28