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Category:Car audioQ:

Number of semisimple ideals in $\mathbb Z_p[\sqrt{ -5}]$

I am having a little trouble solving this question in my coursework. I can see a connection to the number of factors in the prime factorisation of $5$, but I’m not getting anywhere. Any hints would be much appreciated, thanks!

Find the number of semisimple ideals in $\mathbb Z_p[\sqrt{ -5}]$ where $p$ is prime.

A:

Following the advice of the OP, I’m posting what I thought would be helpful to solve this problem.
We want to find the number of semisimple ideals $I$ in the ring $\mathbb{Z}_p[\sqrt{ -5}]$. Let $f(x) = x^2 – 5$. Then the number of semisimple ideals can be found by counting points in $Z(f(x))$

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