If I have an irreducible element $a\in \mathbb{Z}\left [i\right ]$ and $K$ is the prime field of $\mathbb{Z}\left [i\right ]/\left (a\right )$, how do I find $\left [\mathbb{Z}\left [i\right ]/\left (a\right ):K\right ]$?
I tried supposing $a\in \mathbb{Z}$, therefore $a$ is a prime number and $a\equiv 3\pmod 4$. Since $\mathbb{Z}[i]/(a)\cong \mathbb{Z}/a\mathbb{Z}[i]/(a)$ and $K=\mathbb{F}_a$ we have that $[\mathbb{Z}[i]/(a):K]=2$ because $X^2+1$ is irreducible over $\mathbb{F}_a[X]$.
I do not know how to solve the general case.