Express the ideal $(6) \subset \mathbb Z\left[\sqrt {-5}\right]$ as a product of prime ideals.
I know I can write $(6)=(2)(3)=\left(1+\sqrt {-5}\right)\left(1-\sqrt {-5}\right)$. But I guess these factors might not be prime.
What's more, how to solve this kind of problem more systematically?
Best Answer
My answer is shorter than that of Watson, but since mine is informed by more advanced knowledge, you may prefer his answer to mine.
Because $6=2\cdot3$, it’s only necessary to express the ideals $(2)$ and $(3)$ of $R=\Bbb Z\sqrt{-5}$ as products of primes, since we know that prime-ideal decomposition is unique.
I know that the ramified primes are just $2$ and $5$, and because $-5$ is a square modulo $3$, $(3)$ should split. Thus I should find $(2)=\mathfrak p_2^2$ and $(3)=\mathfrak p_3\mathfrak p_3'$. Indeed, \begin{align} (2,1-\sqrt{-5})^2&=(4,2-2\sqrt{-5},-4-2\sqrt{-5})=(4,6,-4-2\sqrt{-5})=(2)\\ (3,1-\sqrt{-5})((3,1+\sqrt{-5})&=(9,3+3\sqrt{-5},3-3\sqrt{-5},-6)=(3)\, \end{align} so that the factorization of $6$ is $\left(2,1-\sqrt{-5}\right)^2\left(3,1-\sqrt{-5}\right)\left(3,1+\sqrt{-5}\right)$.