In the first post regarding this topic, i sketched why an algorithm that counts the number of prime factors of an integer could lead to an algorithm that actually could factorize that integer. The application of such an algorithm to different number fields would lead to non-trivial information about the prime factors due to the definition of prime numbers in $\mathbb{Q}(\sqrt{d})$. The hope is, that this information would eventually allow us to reconstruct the entire prime numbers.
In this post i will show some basic facts about different number field and what information they leak about prime factors of $n$ if $\sigma_0(n)$ is known.
"A number is a mathematical object used to count, label, and measure."