Say L is a number field and its ring of integers is $\mathcal{O}_{L}$. Call a prime ideal I $\subset\mathcal{O}_{L}$ even iff | $\mathcal{O}_{L}$/I | is even iff | $\mathcal{O}_{L}$/I | is a power of 2. [Last iff is due to basic ANT]
For example for L = $\mathbb{Q}[i]$ and $\mathcal{O}_{L}$ = $\mathbb{Z}[i]$ then only even prime ideal is $\langle 1+i \rangle$.
Similarly, I was trying to see what kind of number fields will have how many even prime ideals (or none). Any help would be greatly appreciated.